Case Study / Scenario-Based MCQs for Sub-Topics of Topic 6: Coordinate Geometry

Question 1. Imagine you are looking at a map of a city laid out on a grid. A landmark, the Town Hall, is located at a point where its horizontal position is 4 units to the right of the main vertical road (Y-axis) and its vertical position is 2 units above the main horizontal road (X-axis). Which of the following represents the coordinates of the Town Hall in this grid system?

(A) $(-4, 2)$

(B) $(4, -2)$

(C) $(4, 2)$

(D) $(-4, -2)$

Question 2. A park is located in the region of the city grid where the horizontal coordinates are negative and the vertical coordinates are positive. In which quadrant of the Cartesian plane does the park lie?

(A) First Quadrant

(B) Second Quadrant

(C) Third Quadrant

(D) Fourth Quadrant

Question 3. A library is situated exactly on the main horizontal road (X-axis) of the city grid, 7 units to the left of the intersection of the main roads. What are the coordinates of the library?

(A) $(7, 0)$

(B) $(-7, 0)$

(C) $(0, 7)$

(D) $(0, -7)$

Question 4. A point representing a restaurant on a coordinate map has an abscissa of $-3$ and an ordinate of $-6$. In which quadrant is the restaurant located?

(A) First Quadrant

(B) Second Quadrant

(C) Third Quadrant

(D) Fourth Quadrant

Question 5. On a treasure map, the starting point is marked as the 'Origin'. The first direction says to move to a location where the distance from the horizontal axis is 5 units and the distance from the vertical axis is 3 units. Which of the following could be the coordinates of this location?

(A) $(5, 3)$

(B) $(3, 5)$

(C) $(-3, -5)$

(D) All of the above are possible locations (depending on quadrant).

Question 1. A robot is programmed to start at the origin $(0, 0)$ and move according to given coordinates. Its first instruction is to go to the point $(-5, 2)$. Describe the path the robot should take.

(A) 5 units right, then 2 units up.

(B) 5 units left, then 2 units up.

(C) 5 units right, then 2 units down.

(D) 5 units left, then 2 units down.

Question 2. A grid map shows the location of a well at $(4, 3)$. If you are standing at the origin $(0, 0)$, how would you describe the position of the well?

(A) 4 units down, 3 units right.

(B) 4 units right, 3 units up.

(C) 4 units left, 3 units down.

(D) 4 units up, 3 units left.

Question 3. A teacher asks students to mark the point $(5, 0)$ on a large coordinate grid on the classroom floor. Where should a student stand if the origin is at the center of the floor?

(A) On the vertical axis, 5 units up from the center.

(B) On the horizontal axis, 5 units right of the center.

(C) On the vertical axis, 5 units down from the center.

(D) On the horizontal axis, 5 units left of the center.

Question 4. A surveyor places a marker at a location that is 6 meters to the left of a main north-south line (considered the Y-axis) and 3 meters below a main east-west line (considered the X-axis). If the intersection of the main lines is the origin, what are the coordinates of the marker?

(A) $(6, 3)$

(B) $(-6, 3)$

(C) $(6, -3)$

(D) $(-6, -3)$

Question 5. A child is playing a game on a grid. They are told to move to a point where the absolute value of the x-coordinate is 4 and the point is in the second quadrant. Which point did they move to?

(A) $(4, 2)$

(B) $(-4, -2)$

(C) $(-4, 2)$

(D) $(4, -2)$

Question 1. Two villages, A and B, are located at coordinates $(2, 3)$ and $(8, 11)$ respectively on a map where units are in kilometers. A new road is to be built connecting the two villages directly. What is the length of this new road?

(A) $\sqrt{(8-2)^2 + (11-3)^2}$ km

(B) $\sqrt{6^2 + 8^2}$ km

(C) $\sqrt{36 + 64} = \sqrt{100} = 10$ km

(D) All of the above calculate the correct length.

Question 2. Three friends, Priya, Kiran, and Sneha, live at points P$(1, 5)$, K$(3, 1)$, and S$(5, -3)$ respectively. They want to know if their houses lie on a straight line. Which method using the distance formula can they use to check this?

(A) Check if $PK = KS = PS$.

(B) Check if $PK^2 + KS^2 = PS^2$.

(C) Check if $PK + KS = PS$ (or any permutation of the points).

(D) Calculate the midpoints of PK and KS.

Question 3. A treasure is buried at a point that is equidistant from three landmarks A$(0, 0)$, B$(6, 0)$, and C$(0, 8)$. What is the distance of the treasure point from the origin (landmark A)?

(A) $5$ units (It's the circumcenter of the right triangle ABC, located at the midpoint of the hypotenuse BC. Hypotenuse length $\sqrt{6^2+8^2}=10$. Midpoint is $(3, 4)$. Distance from origin to $(3, 4)$ is $\sqrt{3^2+4^2}=5$.)

(B) $6$ units

(C) $8$ units

(D) $10$ units

Question 4. A carpenter is designing a rectangular table top. The coordinates of three corners are given as $(1, 2)$, $(7, 2)$, and $(7, 5)$. What must be the coordinates of the fourth corner to complete the rectangle, and what are the lengths of the diagonals?

(A) Fourth corner: $(1, 5)$, Diagonal length: $\sqrt{(7-1)^2 + (5-2)^2} = \sqrt{6^2+3^2} = \sqrt{36+9} = \sqrt{45} = 3\sqrt{5}$.

(B) Fourth corner: $(1, 5)$, Diagonal length: $3\sqrt{5}$.

(C) Fourth corner: $(1, 5)$.

(D) All of the above statements are correct.

Question 5. A mobile tower is to be erected at a point on the X-axis such that it is equidistant from two cities located at $(2, -3)$ and $(4, 5)$. What are the coordinates of the optimal tower location?

(A) Let the point on X-axis be $(x, 0)$. Distance from $(x, 0)$ to $(2, -3)$ is $\sqrt{(x-2)^2 + (0-(-3))^2} = \sqrt{(x-2)^2 + 9}$. Distance from $(x, 0)$ to $(4, 5)$ is $\sqrt{(x-4)^2 + (0-5)^2} = \sqrt{(x-4)^2 + 25}$. Equating them: $(x-2)^2 + 9 = (x-4)^2 + 25$. $x^2 - 4x + 4 + 9 = x^2 - 8x + 16 + 25$. $-4x + 13 = -8x + 41$. $4x = 28$. $x = 7$.

(B) $(7, 0)$

(C) $(0, 7)$

(D) $(7, 0)$ is the location.

Question 1. A main highway connects city A at $(2, 5)$ and city B at $(10, 13)$. A rest stop is to be built along the highway, dividing the distance between A and B in the ratio $3:1$ (from A to B). What are the coordinates of the rest stop?

(A) $(\frac{3(10) + 1(2)}{3+1}, \frac{3(13) + 1(5)}{3+1})$

(B) $(\frac{30 + 2}{4}, \frac{39 + 5}{4})$

(C) $(\frac{32}{4}, \frac{44}{4}) = (8, 11)$

(D) All of the above steps correctly calculate the coordinates.

Question 2. A surveyor marks two points P$(4, -1)$ and Q$(-2, -3)$ on a plot of land. They need to locate two points that divide the line segment PQ into three equal parts (trisect it). What are the coordinates of these two points?

(A) The points divide PQ internally in ratios $1:2$ and $2:1$.

(B) Point 1: $(\frac{1(-2) + 2(4)}{1+2}, \frac{1(-3) + 2(-1)}{1+2}) = (\frac{-2+8}{3}, \frac{-3-2}{3}) = (\frac{6}{3}, \frac{-5}{3}) = (2, -5/3)$.

(C) Point 2: $(\frac{2(-2) + 1(4)}{2+1}, \frac{2(-3) + 1(-1)}{2+1}) = (\frac{-4+4}{3}, \frac{-6-1}{3}) = (\frac{0}{3}, \frac{-7}{3}) = (0, -7/3)$.

(D) All of the above steps are correct.

Question 3. A business analyst needs to find the average location of three distribution centers located at $(1, 2)$, $(5, 8)$, and $(6, 4)$. This average location is represented by the centroid of the triangle formed by these points. What are the coordinates of the centroid?

(A) $(\frac{1+5+6}{3}, \frac{2+8+4}{3})$

(B) $(\frac{12}{3}, \frac{14}{3})$

(C) $(4, 14/3)$

(D) All of the above calculate the centroid.

Question 4. A boundary line passes through points A$(5, -2)$ and B$(-1, 4)$. A landmark is located exactly halfway between A and B. What are the coordinates of this landmark?

(A) $(\frac{5+(-1)}{2}, \frac{-2+4}{2})$

(B) $(\frac{4}{2}, \frac{2}{2}) = (2, 1)$

(C) $(2, 1)$

(D) This uses the midpoint formula, which is a special case of the section formula with ratio 1:1.

Question 5. A point P is to be located on the line extending from A$(1, 1)$ to B$(4, 7)$ such that B divides AP in the ratio $3:2$ (meaning AB/BP = 3/2). What are the coordinates of P? (Hint: B divides AP externally in ratio $3:2$, or P divides AB externally in ratio $5:-2$).

(A) Let P divide AB externally in ratio $m:n$. So $m=5$, $n=-2$. P = $(\frac{5(4) + (-2)(1)}{5-2}, \frac{5(7) + (-2)(1)}{5-2}) = (\frac{20-2}{3}, \frac{35-2}{3}) = (\frac{18}{3}, \frac{33}{3}) = (6, 11)$.

(B) $(6, 11)$

(C) $(6, 11)$ are the coordinates.

(D) This uses the section formula for external division.

Question 1. A farmer owns a triangular field with vertices at A$(1, 2)$, B$(7, 2)$, and C$(4, 6)$. He needs to calculate the area of the field to determine how much fertilizer to buy. What is the area of the field?

(A) The base AB lies on the line $y=2$, its length is $|7-1|=6$. The height is the perpendicular distance from C$(4, 6)$ to the line $y=2$, which is $|6-2|=4$.

(B) Area $= \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 4 = 12$ sq units.

(C) Using the coordinate formula: $\frac{1}{2} |1(2-6) + 7(6-2) + 4(2-2)| = \frac{1}{2} |1(-4) + 7(4) + 4(0)| = \frac{1}{2} |-4 + 28| = \frac{1}{2} |24| = 12$ sq units.

(D) All of the above correctly calculate the area.

Question 2. Three cities, P, Q, and R, are located at coordinates $(2, 3), (4, k)$, and $(6, 7)$ respectively. A planner needs to determine if these three cities lie on a single straight railway line. For what value of $k$ would the cities be collinear?

(A) The points are collinear if the area of the triangle formed by them is $0$.

(B) Area $= \frac{1}{2} |2(k-7) + 4(7-3) + 6(3-k)| = 0$.

(C) $|2k - 14 + 4(4) + 18 - 6k| = 0 \implies |2k - 14 + 16 + 18 - 6k| = 0 \implies |-4k + 20| = 0$.

(D) $-4k + 20 = 0 \implies 4k = 20 \implies k = 5$.

Question 3. A landscape architect is designing a garden in the shape of a triangle with vertices at $(0, 0)$, $(5, 0)$, and $(5, 4)$. What is the area of this garden?

(A) $10$ sq units

(B) $20$ sq units

(C) $5$ sq units

(D) $15$ sq units

Question 4. A civil engineer is surveying points $(a, b+c), (b, c+a), (c, a+b)$. They need to confirm if these points form a triangle or lie on a straight line. Using the area formula, they find the area is $0$. What does this imply about the points?

(A) They form a triangle with zero area (impossible for a real triangle).

(B) They are collinear.

(C) They form a degenerate triangle.

(D) Both (B) and (C) are correct interpretations.

Question 5. A student is asked to calculate the area of a region bounded by the points $(0, 0), (4, 0)$, and $(4, 3)$. They use the formula $\frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$. Which calculation is correct?

(A) $\frac{1}{2} |0(0-3) + 4(3-0) + 4(0-0)| = \frac{1}{2} |0 + 12 + 0| = 6$

(B) $\frac{1}{2} |0(0-3) + 4(3-0) + 4(0-0)| = 12$

(C) $\frac{1}{2} |4 \times 3| = 6$ (Using base and height)

(D) Both (A) and (C) are correct calculations.

Question 1. A craftsman is making a triangular wooden sign with vertices at $(0, 0)$, $(6, 0)$, and $(3, 9)$. He wants to hang the sign from a single point so that it balances perfectly. This balancing point is the centroid of the triangle. What are the coordinates of the point where he should hang the sign?

(A) $(\frac{0+6+3}{3}, \frac{0+0+9}{3})$

(B) $(\frac{9}{3}, \frac{9}{3}) = (3, 3)$

(C) $(3, 3)$

(D) This uses the centroid formula.

Question 2. A circular garden is to be designed inside a triangular park such that the garden is tangent to all three sides of the park. The center of this circular garden is the incenter of the triangular park. If the vertices of the park are given, what lines should be found to locate the incenter?

(A) Medians

(B) Altitudes

(C) Angle bisectors

(D) Perpendicular bisectors of the sides

Question 3. A company plans to build a circular pathway that passes through three key locations: a cafeteria at $(1, 1)$, a laboratory at $(5, 1)$, and an administration block at $(3, 5)$. The center of this circular pathway will be the circumcenter of the triangle formed by these locations. What is the property of the circumcenter that makes it the center of the circle passing through the vertices?

(A) It is equidistant from the sides of the triangle.

(B) It is the average of the vertex coordinates.

(C) It is equidistant from the vertices of the triangle.

(D) It is the intersection of the altitudes.

Question 4. An architect is designing the roof of a building in the shape of a triangle. The point where the three supporting beams (altitudes) meet is crucial for the structural stability. This point is known as the orthocenter. If the triangle is right-angled at one vertex, where will the orthocenter be located?

(A) Inside the triangle.

(B) Outside the triangle.

(C) At the vertex where the right angle is located.

(D) At the midpoint of the hypotenuse.

Question 5. For a specific triangle, the centroid, incenter, circumcenter, and orthocenter all coincide. What type of triangle is this?

(A) Scalene triangle

(B) Isosceles triangle

(C) Right-angled triangle

(D) Equilateral triangle

Question 1. A dog is tied to a pole at point A$(0, 0)$ with a rope of fixed length 5 meters. The dog runs around the pole keeping the rope taut. What is the shape of the path traced by the dog, and what is its equation?

(A) Shape: Parabola, Equation: $x^2 = 25y$

(B) Shape: Circle, Equation: $x^2 + y^2 = 25$

(C) Shape: Ellipse, Equation: $\frac{x^2}{25} + \frac{y^2}{25} = 1$

(D) Shape: Circle, Equation: $x^2 + y^2 = 5$

Question 2. A gardener is planting a row of saplings such that each sapling is equally distant from a well located at F$(3, 0)$ and a fence line represented by the line $x = -3$. What is the shape formed by the location of the saplings?

(A) Circle

(B) Ellipse

(C) Parabola

(D) Hyperbola

Question 3. A point P moves such that its distance from point A$(1, 0)$ is equal to its distance from the Y-axis. What is the equation of the locus of P?

(A) Distance from P$(x, y)$ to A$(1, 0)$ is $\sqrt{(x-1)^2 + (y-0)^2}$. Distance from P$(x, y)$ to Y-axis (line $x=0$) is $|x|$.

(B) $\sqrt{(x-1)^2 + y^2} = |x|$. Squaring both sides: $(x-1)^2 + y^2 = x^2$. $x^2 - 2x + 1 + y^2 = x^2$. $y^2 - 2x + 1 = 0$.

(C) $y^2 = 2x - 1$

(D) All of the above lead to the correct equation.

Question 4. A geological survey identifies two points, $F_1$ and $F_2$, from which seismic waves are detected. The difference in the time of arrival of waves from $F_1$ and $F_2$ at a listening station P is constant. Since the speed of the waves is constant, this means the difference in distances $|PF_1 - PF_2|$ is constant. What is the shape of the locus of possible locations of the listening station P?

(A) Circle

(B) Ellipse

(C) Parabola

(D) Hyperbola

Question 5. The set of all points that are equidistant from the lines $y=x$ and $y=-x$ forms a locus. What is this locus?

(A) A single line $y=0$ (X-axis).

(B) A single line $x=0$ (Y-axis).

(C) A pair of lines (the coordinate axes X and Y).

(D) A circle centered at the origin.

Question 1. A company's main office is at the origin $(0, 0)$ of a coordinate system. They open a new branch office at the point $(5, 3)$. To make calculations easier for operations relative to the new branch, they shift the origin to $(5, 3)$. A warehouse was originally located at $(7, 8)$. What are the coordinates of the warehouse in the new coordinate system?

(A) Original coordinates $(x, y) = (7, 8)$. New origin $(h, k) = (5, 3)$. New coordinates $(X, Y) = (x-h, y-k) = (7-5, 8-3) = (2, 5)$.

(B) $(2, 5)$

(C) $(12, 11)$

(D) $(-2, -5)$

Question 2. The equation of a circular boundary in a field is given by $x^2 + y^2 - 10x - 12y + 57 = 0$. To simplify the analysis of this circle, an engineer decides to shift the origin to the center of the circle. What should be the coordinates of the new origin?

(A) The center of $x^2 + y^2 + 2gx + 2fy + c = 0$ is $(-g, -f)$. Here $2g = -10 \implies g = -5$, $2f = -12 \implies f = -6$, $c = 57$. Center is $(-(-5), -(-6)) = (5, 6)$.

(B) $(5, 6)$

(C) $(-5, -6)$

(D) $(10, 12)$

Question 3. A line has the equation $y = 2x + 1$. If the origin is shifted to $(1, 3)$, what is the equation of the line in the new coordinate system $(X, Y)$?

(A) Original equation: $y = 2x + 1$. Shift of origin $(h, k) = (1, 3)$. Use $x = X+h, y = Y+k$. So $x = X+1, y = Y+3$.

(B) Substitute into the original equation: $Y+3 = 2(X+1) + 1$. $Y+3 = 2X + 2 + 1$. $Y+3 = 2X + 3$. $Y = 2X$.

(C) $Y = 2X$

(D) The slope remains the same, and the line passes through the new origin since $(1, 3)$ is on the original line ($3 = 2(1)+1$).

Question 4. An equation $y = x^2 + 4x + 5$ describes a parabolic path. To find the vertex of this parabola easily, we can complete the square: $y = (x^2 + 4x + 4) + 1 = (x+2)^2 + 1$. This is of the form $y-1 = (x+2)^2$. By shifting the origin to the vertex, the equation simplifies. What are the coordinates of the vertex and the required shift?

(A) Vertex is $(-2, 1)$. The shift should be to $(-2, 1)$.

(B) Vertex is $(-2, 1)$.

(C) The equation in the new system would be $Y = X^2$ (using $x = X-2, y = Y+1$).

(D) All of the above statements are correct.

Question 5. After shifting the origin to the point $(h, k)$, the new coordinates of a point P are $(X, Y)$. If the original coordinates of P were $(4, -5)$ and the new coordinates are $(1, -2)$, what was the shift $(h, k)$?

(A) $X = x-h \implies 1 = 4-h \implies h = 3$.

(B) $Y = y-k \implies -2 = -5-k \implies k = -5+2 = -3$.

(C) The shift was to $(3, -3)$.

(D) All of the above steps are correct.

Question 1. A ramp is being built for accessibility. The base of the ramp is at point A$(0, 0)$ and it reaches a height of 3 meters at a horizontal distance of 12 meters, point B$(12, 3)$. What is the slope of the ramp?

(A) Slope = $\frac{\text{change in y}}{\text{change in x}} = \frac{3-0}{12-0} = \frac{3}{12} = 1/4$.

(B) $1/4$

(C) $4$

(D) $1/4$ is the slope.

Question 2. Two roads intersect in a city. The equations of the lines representing the roads are $y = 3x + 2$ and $y = -\frac{1}{3}x + 5$. Are these roads perpendicular to each other?

(A) The slope of the first road is $m_1 = 3$.

(B) The slope of the second road is $m_2 = -1/3$.

(C) Check if $m_1 m_2 = -1$. $3 \times (-1/3) = -1$.

(D) Yes, the roads are perpendicular.

Question 3. A conveyor belt is designed to move objects along a line with equation $2x - 5y = 10$. What is the slope of this conveyor belt's path?

(A) Convert to slope-intercept form: $-5y = -2x + 10 \implies y = \frac{2}{5}x - 2$. Slope is $2/5$.

(B) Using $-A/B$ form: $-2/(-5) = 2/5$.

(C) $2/5$

(D) The slope is $2/5$.

Question 4. Two hiking trails meet. Trail 1 has a slope of $1/2$. Trail 2 makes an angle of $45^\circ$ with Trail 1. What are the possible slopes of Trail 2?

(A) Using $\tan\theta = |\frac{m_2 - m_1}{1 + m_1 m_2}|$, with $\theta = 45^\circ$, $\tan 45^\circ = 1$, $m_1 = 1/2$. $1 = |\frac{m_2 - 1/2}{1 + (1/2)m_2}| = |\frac{2m_2 - 1}{2 + m_2}|$.

(B) $\frac{2m_2 - 1}{2 + m_2} = 1 \implies 2m_2 - 1 = 2 + m_2 \implies m_2 = 3$.

(C) $\frac{2m_2 - 1}{2 + m_2} = -1 \implies 2m_2 - 1 = -2 - m_2 \implies 3m_2 = -1 \implies m_2 = -1/3$.

(D) Possible slopes are $3$ and $-1/3$.

Question 5. A section of a wall is vertical. What is the slope of the line representing this section of the wall in a coordinate system where the ground is the X-axis?

(A) $0$

(B) $1$

(C) $-1$

(D) Undefined

Question 1. A fence is to be built along a straight line connecting two points, P$(2, 3)$ and Q$(5, 9)$. What is the equation of the line representing the fence?

(A) Slope $m = \frac{9-3}{5-2} = \frac{6}{3} = 2$.

(B) Using point-slope form with P$(2, 3)$: $y - 3 = 2(x - 2) \implies y - 3 = 2x - 4 \implies y = 2x - 1$.

(C) Using two-point form: $\frac{y-3}{x-2} = \frac{9-3}{5-2} \implies \frac{y-3}{x-2} = 2 \implies y-3 = 2(x-2) \implies y = 2x - 1$.

(D) All of the above correctly find the equation.

Question 2. A road passes through the point $(1, -1)$ and has a slope of $-1$. What is the equation of the line representing this road?

(A) $y = -x - 1$

(B) $y + 1 = -1(x - 1) \implies y + 1 = -x + 1 \implies y = -x$.

(C) $y = x$

(D) $y = -x$ is the equation.

Question 3. A company's budget line shows that they spend $\textsf{₹}1000$ on two items, X and Y. If item X costs $\textsf{₹}100$ each and item Y costs $\textsf{₹}250$ each, and they buy $x$ units of X and $y$ units of Y, the equation is $100x + 250y = 1000$. What is the intercept form of this budget line?

(A) $100x + 250y = 1000$. Divide by 1000: $\frac{100x}{1000} + \frac{250y}{1000} = 1$. $\frac{x}{10} + \frac{y}{4} = 1$.

(B) $\frac{x}{10} + \frac{y}{4} = 1$

(C) The x-intercept is 10 (buy 10 units of X, 0 of Y). The y-intercept is 4 (buy 4 units of Y, 0 of X).

(D) All of the above are correct.

Question 4. A straight pipeline is to be laid from a source along a line whose perpendicular distance from a reference point (origin) is 5 meters, and the perpendicular from the origin to the pipeline makes an angle of $30^\circ$ with the positive X-axis. What is the equation of the line representing the pipeline in normal form?

(A) $x \cos 30^\circ + y \sin 30^\circ = 5$

(B) $x \frac{\sqrt{3}}{2} + y \frac{1}{2} = 5$

(C) $\sqrt{3}x + y = 10$

(D) All of the above represent the equation.

Question 5. An equation $y = -4x + 7$ describes the relationship between two variables in an experiment. What does the value 7 represent in the context of the graph of this equation?

(A) The slope of the line.

(B) The x-intercept (where the line crosses the X-axis).

(C) The y-intercept (where the line crosses the Y-axis).

(D) The perpendicular distance from the origin.

Question 1. A company uses the equation $5x + 2y - 20 = 0$ to represent a production constraint. What is the slope of this line?

(A) $5/2$

(B) $-5/2$

(C) $2/5$

(D) $-2/5$

Question 2. Two supply routes are represented by the lines $x + 2y = 5$ and $3x - y = 1$. If supplies are delivered from the intersection point of these two routes, what is the location of the delivery point?

(A) Solve the system of equations: $x + 2y = 5$, $3x - y = 1$. Multiply second equation by 2: $6x - 2y = 2$. Add to first: $(x + 2y) + (6x - 2y) = 5 + 2 \implies 7x = 7 \implies x = 1$. Substitute $x=1$ into $x+2y=5$: $1+2y=5 \implies 2y=4 \implies y=2$.

(B) $(1, 2)$

(C) $(2, 1)$

(D) $(1, 2)$ is the intersection point.

Question 3. A line is given by the general equation $4x - 3y + 12 = 0$. Convert this to the intercept form $\frac{x}{a} + \frac{y}{b} = 1$. What are the x and y intercepts?

(A) $4x - 3y = -12$. Divide by -12: $\frac{4x}{-12} - \frac{3y}{-12} = \frac{-12}{-12}$. $\frac{x}{-3} + \frac{y}{4} = 1$.

(B) X-intercept $a = -3$, Y-intercept $b = 4$.

(C) X-intercept is $-3$.

(D) Y-intercept is $4$.

Question 4. Two production lines follow paths given by $2x + 5y = 10$ and $4x + 10y = 25$. Are these lines parallel, intersecting, or coincident?

(A) Compare ratios of coefficients: $a_1/a_2 = 2/4 = 1/2$. $b_1/b_2 = 5/10 = 1/2$. $c_1/c_2 = -10/-25 = 10/25 = 2/5$.

(B) $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$.

(C) The lines are parallel and distinct.

(D) The lines are parallel.

Question 5. A linear cost function is given by $C = 500 + 10x$, where $C$ is the total cost and $x$ is the number of items produced. Represent this as a general linear equation in terms of $x$ and $C$.

(A) $10x - C + 500 = 0$

(B) $10x - C + 500 = 0$ is a general linear equation in variables $x$ and $C$.

(C) $C - 10x - 500 = 0$

(D) Both (A) and (C) are equivalent general linear equations.

Question 1. A building is located at point P$(2, 1)$. A straight road is represented by the equation $4x + 3y - 12 = 0$. What is the shortest distance from the building to the road?

(A) Use the distance formula from a point to a line: $\frac{|Ax_0 + By_0 + C|}{\sqrt{A^2+B^2}}$. Here $(x_0, y_0) = (2, 1)$, $A=4, B=3, C=-12$.

(B) Distance = $\frac{|4(2) + 3(1) - 12|}{\sqrt{4^2+3^2}} = \frac{|8 + 3 - 12|}{\sqrt{16+9}} = \frac{|-1|}{\sqrt{25}} = \frac{1}{5}$.

(C) $1/5$ unit.

(D) All of the above steps are correct.

Question 2. Two parallel railway tracks are represented by the equations $y = 2x + 3$ and $y = 2x - 7$. What is the distance between the tracks?

(A) Rewrite in general form: $2x - y + 3 = 0$ and $2x - y - 7 = 0$. $A=2, B=-1, C_1=3, C_2=-7$.

(B) Distance = $\frac{|C_1 - C_2|}{\sqrt{A^2+B^2}} = \frac{|3 - (-7)|}{\sqrt{2^2+(-1)^2}} = \frac{|10|}{\sqrt{4+1}} = \frac{10}{\sqrt{5}}$.

(C) $\frac{10}{\sqrt{5}} = \frac{10\sqrt{5}}{5} = 2\sqrt{5}$ units.

(D) All of the above steps are correct.

Question 3. A town plans to build a new road that must pass through the intersection of two existing main roads, represented by the lines $x + y = 4$ and $x - y = 2$. The new road must also pass through a landmark located at $(5, 0)$. What is the equation of the new road?

(A) The family of lines passing through the intersection of $x+y-4=0$ and $x-y-2=0$ is $(x+y-4) + \lambda(x-y-2) = 0$.

(B) The line passes through $(5, 0)$. Substitute $(5, 0)$: $(5+0-4) + \lambda(5-0-2) = 0 \implies 1 + \lambda(3) = 0 \implies 3\lambda = -1 \implies \lambda = -1/3$.

(C) Substitute $\lambda = -1/3$ back into the family equation: $(x+y-4) - \frac{1}{3}(x-y-2) = 0$. Multiply by 3: $3(x+y-4) - (x-y-2) = 0$. $3x+3y-12 - x+y+2 = 0$. $2x + 4y - 10 = 0$. Divide by 2: $x + 2y - 5 = 0$.

(D) The equation of the new road is $x + 2y - 5 = 0$.

Question 4. A company wants to find the most central location for a small office that serves two nearby towns. The towns are located such that they define a line segment, and the office should be along the main road connecting them. If the towns are at $(2, 3)$ and $(8, 11)$, and the company wants the office to be twice as far from $(8, 11)$ as it is from $(2, 3)$ along the road (ratio 1:2 from (2,3) to (8,11)), what are the coordinates of the office?

(A) This is an internal division problem with ratio 1:2 from $(2, 3)$ to $(8, 11)$.

(B) Coordinates = $(\frac{1(8) + 2(2)}{1+2}, \frac{1(11) + 2(3)}{1+2}) = (\frac{8+4}{3}, \frac{11+6}{3}) = (\frac{12}{3}, \frac{17}{3}) = (4, 17/3)$.

(C) $(4, 17/3)$

(D) This uses the section formula for internal division in 3D.

Question 5. A point P is on the line $y = 2x + 1$. What is the distance of P from the line $2x - y + 5 = 0$? Note that the given point P is on the line $2x-y+1=0$, which is parallel to the second line. The distance should be constant for any point on the first line.

(A) The lines are $2x - y + 1 = 0$ and $2x - y + 5 = 0$. They are parallel.

(B) Distance between them is $\frac{|C_1 - C_2|}{\sqrt{A^2+B^2}} = \frac{|1 - 5|}{\sqrt{2^2+(-1)^2}} = \frac{|-4|}{\sqrt{5}} = \frac{4}{\sqrt{5}}$.

(C) $4/\sqrt{5}$ units.

(D) Any point on the first line is $(x_0, 2x_0+1)$. Distance from $(x_0, 2x_0+1)$ to $2x-y+5=0$ is $\frac{|2x_0 - (2x_0+1) + 5|}{\sqrt{2^2+(-1)^2}} = \frac{|2x_0 - 2x_0 - 1 + 5|}{\sqrt{5}} = \frac{|4|}{\sqrt{5}} = 4/\sqrt{5}$.

Question 1. A drone is hovering at a point with coordinates $(3, 4, 5)$ meters relative to a base station at the origin. A horizontal sensor is located on the XY-plane. What is the altitude of the drone above this sensor plane?

(A) 3 meters

(B) 4 meters

(C) 5 meters

(D) $\sqrt{3^2+4^2}$ meters

Question 2. An object is located at point P$( -2, -3, 1)$. In which octant of the 3D Cartesian system is the object located?

(A) Octant I ($+,+,+$)

(B) Octant II ($-,+,+$)

(C) Octant III ($-, -,+$)

(D) Octant VI ($-,-,-$)

Question 3. A laser pointer is directed along the Z-axis. A dust particle is floating in the air at point P$(2, 5, 0)$. Where is the dust particle located relative to the Z-axis?

(A) On the Z-axis.

(B) On the XY-plane.

(C) On the YZ-plane.

(D) On the XZ-plane.

Question 4. A pressure sensor is placed on a wall which corresponds to the XZ-plane in a room. What is the y-coordinate of the location of this sensor?

(A) It must be 0.

(B) It must be positive.

(C) It must be negative.

(D) It can be any real number.

Question 5. In a manufacturing unit, three conveyor belts run along the X, Y, and Z axes, meeting at a central point. This central point serves as the reference point for tracking items. What are the coordinates of this reference point?

(A) $(1, 1, 1)$

(B) $(0, 0, 0)$

(C) $(1, 0, 0)$

(D) $(0, 1, 0)$

Question 1. Two satellites are orbiting Earth, with their positions given by coordinates A$(100, 200, 300)$ km and B$(500, 600, 1500)$ km relative to a tracking station at the origin. What is the distance between the two satellites?

(A) $\sqrt{(500-100)^2 + (600-200)^2 + (1500-300)^2}$ km

(B) $\sqrt{400^2 + 400^2 + 1200^2}$ km

(C) $\sqrt{160000 + 160000 + 1440000} = \sqrt{1760000} = \sqrt{160000 \times 11} = 400\sqrt{11}$ km

(D) All of the above calculate the correct distance.

Question 2. A sensor is placed at point P$(2, -3, 4)$ in a room. What is the distance of this sensor from the ceiling, which is represented by the plane $z=10$? (Assuming the floor is $z=0$).

(A) $|10 - 4| = 6$ units.

(B) $\sqrt{(2-2)^2 + (-3-(-3))^2 + (10-4)^2}$

(C) 6 units.

(D) This is the perpendicular distance from the point to the plane $z=10$.

Question 3. Three markers are placed at points A$(1, 0, 0)$, B$(0, 1, 0)$, and C$(0, 0, 1)$. Are these three markers collinear?

(A) Calculate distances AB, BC, AC.

(B) AB = $\sqrt{(-1)^2+1^2+0^2} = \sqrt{2}$. BC = $\sqrt{0^2+(-1)^2+1^2} = \sqrt{2}$. AC = $\sqrt{(-1)^2+0^2+1^2} = \sqrt{2}$.

(C) Since AB = BC = AC = $\sqrt{2}$, the points form an equilateral triangle. Equilateral triangle vertices are not collinear (unless degenerate, which these aren't).

(D) No, the points are not collinear.

Question 4. Find the point on the X-axis that is equidistant from points P$(1, 2, -1)$ and Q$(-2, 1, 3)$.

(A) Let the point on the X-axis be $(x, 0, 0)$. Distance from $(x, 0, 0)$ to P is $\sqrt{(x-1)^2 + (0-2)^2 + (0-(-1))^2} = \sqrt{(x-1)^2 + 4 + 1} = \sqrt{(x-1)^2 + 5}$. Distance from $(x, 0, 0)$ to Q is $\sqrt{(x-(-2))^2 + (0-1)^2 + (0-3)^2} = \sqrt{(x+2)^2 + 1 + 9} = \sqrt{(x+2)^2 + 10}$. Equating: $(x-1)^2 + 5 = (x+2)^2 + 10$. $x^2 - 2x + 1 + 5 = x^2 + 4x + 4 + 10$. $-2x + 6 = 4x + 14$. $-8 = 6x$. $x = -8/6 = -4/3$.

(B) $(-4/3, 0, 0)$

(C) $(-4/3, 0, 0)$ is the point.

(D) This problem uses the distance formula in 3D.

Question 5. A robot arm has its base at the origin $(0, 0, 0)$. The end effector needs to reach a target point at $(3, -4, 12)$. What is the minimum required length of the robotic arm (assuming it can extend in a straight line from the origin)?

(A) The minimum length is the distance from the origin to the target point.

(B) Distance = $\sqrt{3^2 + (-4)^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13$ units.

(C) 13 units.

(D) This is calculated using the distance formula from the origin in 3D.

Question 1. A straight support beam is placed between two points A$(1, -2, 3)$ and B$(7, 4, 9)$. A connection point needs to be fixed exactly at the midpoint of the beam. What are the coordinates of this connection point?

(A) $(\frac{1+7}{2}, \frac{-2+4}{2}, \frac{3+9}{2})$

(B) $(\frac{8}{2}, \frac{2}{2}, \frac{12}{2}) = (4, 1, 6)$

(C) $(4, 1, 6)$

(D) This uses the midpoint formula in 3D.

Question 2. A company has warehouses at locations P$(2, 3, 4)$ and Q$(8, 0, 10)$. A new distribution center is to be built along the straight line connecting P and Q, such that it is closer to P, dividing the distance from P to Q in the ratio $1:2$. What are the coordinates of the distribution center?

(A) This is an internal division problem with ratio 1:2 from P to Q.

(B) Coordinates = $(\frac{1(8) + 2(2)}{1+2}, \frac{1(0) + 2(3)}{1+2}, \frac{1(10) + 2(4)}{1+2}) = (\frac{8+4}{3}, \frac{0+6}{3}, \frac{10+8}{3}) = (\frac{12}{3}, \frac{6}{3}, \frac{18}{3}) = (4, 2, 6)$.

(C) $(4, 2, 6)$

(D) This uses the section formula for internal division in 3D.

Question 3. A point R is located on the line containing segment AB, but outside the segment, such that R divides AB externally in the ratio $3:1$. If A is at $(1, 1, 1)$ and B is at $(2, 3, 4)$, what are the coordinates of R?

(A) R divides AB externally in ratio $m:n = 3:1$. A=$(x_1, y_1, z_1)=(1,1,1)$, B=$(x_2, y_2, z_2)=(2,3,4)$.

(B) Coordinates of R = $(\frac{m x_2 - n x_1}{m-n}, \frac{m y_2 - n y_1}{m-n}, \frac{m z_2 - n z_1}{m-n})$

(C) R = $(\frac{3(2) - 1(1)}{3-1}, \frac{3(3) - 1(1)}{3-1}, \frac{3(4) - 1(1)}{3-1}) = (\frac{6-1}{2}, \frac{9-1}{2}, \frac{12-1}{2}) = (\frac{5}{2}, \frac{8}{2}, \frac{11}{2}) = (2.5, 4, 5.5)$.

(D) $(2.5, 4, 5.5)$

Question 4. A company has three major production units located at $(0, 0, 0)$, $(6, 0, 0)$, and $(3, 9, 0)$. They want to locate a central coordination office that represents the "average" position of these units in 3D space (assuming they form a triangle on the XY plane and their z-coordinates are 0). This is the centroid of the triangle. What are the coordinates of the coordination office?

(A) $(\frac{0+6+3}{3}, \frac{0+0+9}{3}, \frac{0+0+0}{3})$

(B) $(\frac{9}{3}, \frac{9}{3}, 0) = (3, 3, 0)$

(C) $(3, 3, 0)$

(D) This uses the centroid formula for a triangle in 3D.

Question 5. A construction team is placing anchors along a line segment between two points A$(1, 2, 3)$ and B$(7, 8, 9)$. They need to place anchors at points that divide the segment AB into three equal parts. What are the coordinates of the anchor point closest to B?

(A) This point divides AB internally in the ratio $2:1$ (from A to B).

(B) Coordinates = $(\frac{2(7) + 1(1)}{2+1}, \frac{2(8) + 1(2)}{2+1}, \frac{2(9) + 1(3)}{2+1}) = (\frac{14+1}{3}, \frac{16+2}{3}, \frac{18+3}{3}) = (\frac{15}{3}, \frac{18}{3}, \frac{21}{3}) = (5, 6, 7)$.

(C) $(5, 6, 7)$

(D) This uses the section formula for internal division in 3D.

Question 1. A flashlight beam is directed towards a wall. If the wall is flat and the beam is shaped like a cone, what shapes might the edge of the illuminated area form on the wall?

(A) Circle, Ellipse, Parabola, Hyperbola

(B) Only Circle and Ellipse

(C) Only Parabola and Hyperbola

(D) Only Circle and Parabola

Question 2. A satellite dish is designed to collect signals from a distant satellite. The shape of the dish is typically a part of a conic section that reflects parallel incoming rays to a single point (the receiver). What conic section is used for the shape of the dish?

(A) Circle

(B) Ellipse

(C) Parabola

(D) Hyperbola

Question 3. The orbit of a planet around a star, according to Kepler's laws, is an ellipse with the star at one of the foci. This shape is characterized by the sum of the distances from the planet to the two foci being constant. What is the range of eccentricity for such an orbit?

(A) $e = 0$

(B) $0 < e < 1$

(C) $e = 1$

(D) $e > 1$

Question 4. A mirror is shaped like a hyperbola. A property of this shape is that a light ray directed towards one focus is reflected towards the other focus. What is the defining property of a hyperbola in terms of distances from two foci?

(A) The sum of the distances is constant.

(B) The difference of the distances is constant.

(C) The ratio of the distances is constant.

(D) The product of the distances is constant.

Question 5. A stone is thrown upwards and outwards. Neglecting air resistance, the path it follows is a type of conic section. What is this shape?

(A) Circle

(B) Ellipse (part of)

(C) Parabola (part of)

(D) Hyperbola (part of)

Question 1. A circular park boundary is represented by the equation $(x-5)^2 + (y+3)^2 = 100$. Where is the center of this park located, and what is the radius of the boundary?

(A) Center: $(5, 3)$, Radius: 10

(B) Center: $(-5, -3)$, Radius: 100

(C) Center: $(5, -3)$, Radius: 10

(D) Center: $(-5, 3)$, Radius: 100

Question 2. A radio tower is placed at the origin $(0, 0)$. Its signal can reach up to a distance of 15 km. What equation represents the boundary of the area covered by the signal?

(A) $x + y = 15$

(B) $x^2 + y^2 = 15$

(C) $x^2 + y^2 = 225$

(D) $(x-15)^2 + (y-15)^2 = 0$

Question 3. A straight road is represented by the line $y = x + 1$. A circular fountain is located with its center at $(0, 0)$ and radius 2. Does the road intersect the fountain, is it tangent to it, or does it miss the fountain?

(A) Distance from $(0, 0)$ to $x - y + 1 = 0$ is $\frac{|1|}{\sqrt{1^2+(-1)^2}} = \frac{1}{\sqrt{2}}$. Radius is 2. Since $1/\sqrt{2} \approx 0.707 < 2$, the road intersects the fountain at two points.

(B) The road is tangent to the fountain.

(C) The road intersects the fountain at two points.

(D) The road misses the fountain.

Question 4. Two circular gears touch each other externally. Gear 1 has center at $(2, 0)$ and radius 3. Gear 2 has center at $(k, 0)$ and radius 5. What is the value of $k$ if they touch externally?

(A) The distance between centers must equal the sum of radii: $|k-2| = 3+5 = 8$.

(B) $k-2 = 8$ or $k-2 = -8$.

(C) $k = 10$ or $k = -6$.

(D) All of the above steps are correct.

Question 5. An earthquake's epicenter is at a certain location. Seismograph stations detect the quake. If a station at $(x_1, y_1)$ detects the seismic waves, and the wave speed is constant, the possible locations of the epicenter form a circle around $(x_1, y_1)$. If three stations at $(0, 0), (6, 0), (0, 8)$ detect the quake simultaneously (meaning they are equidistant from the epicenter), what is the location of the epicenter?

(A) The epicenter is the circumcenter of the triangle formed by the stations.

(B) The triangle formed by the stations is a right triangle at $(0, 0)$.

(C) The circumcenter of a right triangle is the midpoint of the hypotenuse. Hypotenuse connects $(6, 0)$ and $(0, 8)$. Midpoint is $(\frac{6+0}{2}, \frac{0+8}{2}) = (3, 4)$.

(D) The epicenter is at $(3, 4)$.

Question 1. A parabolic reflector is designed for a spotlight. The light bulb should be placed at the focus of the parabola to ensure the light rays are reflected as a parallel beam. If the equation of the parabolic cross-section is $y^2 = 16x$ and the vertex is at the origin, where should the light bulb be placed?

(A) The equation is $y^2 = 4ax$. $4a = 16 \implies a = 4$.

(B) The focus is at $(a, 0)$ for this form of parabola.

(C) The focus is at $(4, 0)$.

(D) The light bulb should be placed at $(4, 0)$.

Question 2. The cable of a suspension bridge hangs in a shape that is approximately a parabola. If the lowest point of the cable is at $(0, 0)$ and the towers are at $(100, 50)$ and $(-100, 50)$ (in meters), what is the equation of the parabola?

(A) The vertex is at $(0, 0)$. The parabola opens upwards and is symmetric about the Y-axis. The equation is of the form $x^2 = 4ay$.

(B) The point $(100, 50)$ lies on the parabola. Substitute into the equation: $100^2 = 4a(50)$. $10000 = 200a$. $a = 10000/200 = 50$.

(C) The equation is $x^2 = 4(50)y \implies x^2 = 200y$.

(D) All of the above are correct.

Question 3. A projectile is launched from the origin $(0, 0)$ following a parabolic path $y = x - \frac{x^2}{100}$. What is the maximum height reached by the projectile?

(A) Rewrite the equation by completing the square for x: $y = -( \frac{x^2}{100} - x) = - \frac{1}{100}(x^2 - 100x) = - \frac{1}{100}(x^2 - 100x + 50^2 - 50^2) = - \frac{1}{100}((x-50)^2 - 2500) = - \frac{1}{100}(x-50)^2 + 25$.

(B) The equation is $y - 25 = - \frac{1}{100}(x-50)^2$, which is $(x-50)^2 = -100(y-25)$.

(C) This is a parabola opening downwards with vertex at $(50, 25)$.

(D) The maximum height is the y-coordinate of the vertex, which is 25 units.

Question 4. A parabolic archway has its vertex at the origin and opens downwards. If the arch is 10 meters wide at the base and 5 meters high, what is the equation of the parabola?

(A) The vertex is $(0, 0)$. The parabola opens downwards, so the equation is $x^2 = -4ay$.

(B) The width of the base is 10m, and height is 5m. The points at the base are $(-5, -5)$ and $(5, -5)$.

(C) Substitute $(5, -5)$ into $x^2 = -4ay$: $5^2 = -4a(-5) \implies 25 = 20a \implies a = 25/20 = 5/4$.

(D) The equation is $x^2 = -4(5/4)y \implies x^2 = -5y$.

Question 5. The reflective property of a parabola is used in designing searchlights. If the parabolic reflector has the equation $y^2 = 4x$ and the vertex is at the origin, where should the light source be placed?

(A) At the vertex.

(B) At the focus.

(C) At the directrix.

(D) At the endpoint of the latus rectum.

Question 1. An elliptical running track is to be designed. The length of the major axis is 200 meters and the length of the minor axis is 160 meters. If the center of the track is at the origin and the major axis is along the X-axis, what is the equation of the ellipse?

(A) Length of major axis $2a = 200 \implies a = 100$. Length of minor axis $2b = 160 \implies b = 80$. Major axis along X-axis, center at origin: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

(B) $\frac{x^2}{100^2} + \frac{y^2}{80^2} = 1 \implies \frac{x^2}{10000} + \frac{y^2}{6400} = 1$.

(C) $\frac{x^2}{10000} + \frac{y^2}{6400} = 1$.

(D) All of the above are correct steps.

Question 2. A 'whispering gallery' is an elliptical room where a person whispering at one focus can be clearly heard at the other focus. If such a room is an ellipse with equation $\frac{x^2}{400} + \frac{y^2}{300} = 1$ (units in feet), how far apart are the two foci where speakers/listeners should stand?

(A) The equation is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. $a^2 = 400 \implies a = 20$. $b^2 = 300 \implies b = \sqrt{300} = 10\sqrt{3}$. Since $a^2 > b^2$, the major axis is along the X-axis. $c^2 = a^2 - b^2 = 400 - 300 = 100 \implies c = 10$.

(B) The foci are at $(\pm c, 0) = (\pm 10, 0)$.

(C) The distance between the two foci is $2c = 2(10) = 20$ feet.

(D) All of the above are correct steps.

Question 3. The orbit of a comet around the sun is an ellipse. If the equation of the orbit is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the sun is at a focus, what is the eccentricity of the orbit, and what does it tell us about the shape?

(A) Eccentricity $e = c/a$ or $c/b$. For an ellipse, $0 \le e < 1$.

(B) A smaller eccentricity means the orbit is more circular.

(C) An eccentricity close to 1 means the orbit is very elongated.

(D) All of the above are correct statements about eccentricity and shape.

Question 4. An archway is in the shape of a semi-ellipse. The base of the arch is 10 meters wide and the height at the center is 4 meters. If the base is along the X-axis and the center of the base is at the origin, what is the equation of the full ellipse?

(A) The width of the base is $2a = 10 \implies a = 5$. The height at the center is $b = 4$. Since the base is along the X-axis, $a$ is the semi-major axis if $a>b$, or $b$ is the semi-minor axis if $a>b$. Here $a=5, b=4$, so major axis is along X. Equation is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

(B) $\frac{x^2}{5^2} + \frac{y^2}{4^2} = 1 \implies \frac{x^2}{25} + \frac{y^2}{16} = 1$.

(C) $\frac{x^2}{25} + \frac{y^2}{16} = 1$.

(D) All of the above are correct steps.

Question 5. An ellipse is defined by the parametric equations $x = 3\cos t, y = 5\sin t$. What is the length of the major axis of this ellipse?

(A) Eliminate t: $\frac{x}{3} = \cos t, \frac{y}{5} = \sin t$. $\frac{x^2}{9} + \frac{y^2}{25} = \cos^2 t + \sin^2 t = 1$.

(B) The equation is $\frac{x^2}{9} + \frac{y^2}{25} = 1$. Here $a^2 = 9 \implies a = 3$, $b^2 = 25 \implies b = 5$. Since $b > a$, the major axis is along the Y-axis, and its length is $2b$.

(C) Length of major axis $= 2 \times 5 = 10$.

(D) All of the above are correct steps.

Question 1. Two detecting stations, A and B, are located 10 km apart. An event (like an explosion) occurs at point P. The sound is detected at A 3 seconds before it is detected at B. If the speed of sound is 340 m/s (0.34 km/s), the difference in distances $|PA - PB|$ is constant. This means P lies on a hyperbola. What is the value of $2a$ for this hyperbola?

(A) $|PA - PB| = \text{speed} \times \text{time difference} = 0.34 \text{ km/s} \times 3 \text{ s} = 1.02 \text{ km}$.

(B) By definition, $|PA - PB| = 2a$.

(C) $2a = 1.02$ km.

(D) All of the above are correct.

Question 2. A comet follows a hyperbolic path around the sun, passing very close to it. The sun is at one focus. If the equation of the hyperbola is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, which axis contains the sun?

(A) X-axis (the transverse axis, containing the foci).

(B) Y-axis (the conjugate axis).

(C) Either axis depending on the value of $a$ and $b$.

(D) Neither axis.

Question 3. The cooling tower of a power plant often has a hyperbolic cross-section. Suppose the cross-section is given by $\frac{x^2}{36} - \frac{y^2}{64} = 1$ (units in meters). What are the equations of the asymptotes of this hyperbola?

(A) The equation is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. $a^2 = 36 \implies a = 6$. $b^2 = 64 \implies b = 8$.

(B) The asymptotes are $y = \pm \frac{b}{a}x$.

(C) $y = \pm \frac{8}{6}x = \pm \frac{4}{3}x$.

(D) All of the above are correct steps.

Question 4. A hyperbola has the equation $\frac{y^2}{100} - \frac{x^2}{25} = 1$. What is the eccentricity of this hyperbola?

(A) The equation is $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$. $a^2 = 100 \implies a = 10$. $b^2 = 25 \implies b = 5$. $c^2 = a^2 + b^2 = 100 + 25 = 125$. $c = \sqrt{125} = 5\sqrt{5}$.

(B) Eccentricity $e = c/a = 5\sqrt{5}/10 = \sqrt{5}/2$.

(C) $e = \sqrt{1 + b^2/a^2} = \sqrt{1 + 25/100} = \sqrt{1 + 1/4} = \sqrt{5/4} = \sqrt{5}/2$.

(D) All of the above are correct steps.

Question 5. The reflective property of a hyperbola states that a ray aimed at one focus is reflected as if it came from the other focus. This property is used in some telescope designs (Cassegrain telescopes). What is the defining characteristic of a hyperbola in terms of its foci?

(A) The sum of distances to the foci is constant.

(B) The difference of distances to the foci is constant.

(C) The ratio of distances to the foci is constant (eccentricity $e>1$).

(D) Both (B) and (C) are fundamental properties, with the eccentricity definition $PF = e \cdot PD$ linking distance to focus and directrix.

Question 1. A satellite orbits Earth in a perfectly circular path with radius 8000 km, centered at the origin. Its position $(x, y)$ at time $t$ (in hours) can be described parametrically, say starting on the positive X-axis and moving counterclockwise. If its speed is such that it completes one orbit in 24 hours, which of the following parametric equations could describe its path?

(A) $x = 8000t, y = 8000t$

(B) $x = 8000\cos(\frac{2\pi t}{24}), y = 8000\sin(\frac{2\pi t}{24})$

(C) $x = 8000t^2, y = 16000t$

(D) $x = 8000\sec(\frac{2\pi t}{24}), y = 8000\tan(\frac{2\pi t}{24})$

Question 2. The trajectory of a drone is described by the parametric equations $x = 5t$, $y = -2t^2 + 10t$, where $t$ is time in seconds and $x, y$ are in meters. What type of path does the drone follow in the XY plane?

(A) Eliminate t: $t = x/5$. Substitute into the y equation: $y = -2(x/5)^2 + 10(x/5) = -2x^2/25 + 2x$. This is a quadratic equation of the form $y = Ax^2 + Bx$, which represents a parabola.

(B) Circle

(C) Ellipse

(D) Parabola

Question 3. A point moves according to the parametric equations $x = 2 + 3\cos\theta$, $y = 1 + 3\sin\theta$. What conic section does this represent?

(A) Rewrite: $x-2 = 3\cos\theta, y-1 = 3\sin\theta$. Square and add: $(x-2)^2 + (y-1)^2 = (3\cos\theta)^2 + (3\sin\theta)^2 = 9(\cos^2\theta + \sin^2\theta) = 9$.

(B) $(x-2)^2 + (y-1)^2 = 9$. This is the equation of a circle centered at $(2, 1)$ with radius 3.

(C) Parabola

(D) Circle

Question 4. Parametric equations are useful for analyzing the velocity and acceleration of a point moving along a curve. If a particle's position is given by $x(t) = t^2, y(t) = t^3 - t$, what is its velocity vector at time $t=1$?

(A) Velocity vector is $(dx/dt, dy/dt)$. $dx/dt = 2t$, $dy/dt = 3t^2 - 1$.

(B) At $t=1$: $dx/dt |_{t=1} = 2(1) = 2$. $dy/dt |_{t=1} = 3(1)^2 - 1 = 3-1 = 2$.

(C) Velocity vector at $t=1$ is $(2, 2)$.

(D) All of the above are correct steps.

Question 5. The shape of an object's boundary is given by the parametric equations $x = 5\cosh t, y = 3\sinh t$. What type of conic section is this?

(A) Use hyperbolic identity $\cosh^2 t - \sinh^2 t = 1$. $\frac{x}{5} = \cosh t, \frac{y}{3} = \sinh t$. $(\frac{x}{5})^2 - (\frac{y}{3})^2 = \cosh^2 t - \sinh^2 t = 1$. $\frac{x^2}{25} - \frac{y^2}{9} = 1$.

(B) This is the equation of a hyperbola centered at the origin with transverse axis along the X-axis.

(C) Parabola

(D) Hyperbola

Question 1. In urban planning, coordinate geometry is used extensively. Which of the following are typical applications?

(A) Mapping locations of buildings, roads, and utilities.

(B) Calculating distances between points of interest.

(C) Designing the layout of new neighborhoods or parks.

(D) All of the above.

Question 2. An engineer uses coordinate geometry to design a bridge structure. Which geometric concepts are likely to be applied?

(A) Equations of lines to represent beams and supports.

(B) Distance formula to calculate lengths of components.

(C) Angles between lines to ensure structural integrity.

(D) All of the above.

Question 3. A computer graphics programmer is designing a 3D model of a car. How is coordinate geometry relevant here?

(A) Representing the position of each point on the car's surface using 3D coordinates.

(B) Using equations of surfaces (like spheres, cylinders, planes) to define parts of the car.

(C) Applying transformations (translation, rotation, scaling) to move and orient the car model in a virtual scene.

(D) All of the above.

Question 4. In physics, coordinate geometry is crucial for describing motion. If the position of a particle at time $t$ is given by $(x(t), y(t))$, which coordinate geometry concepts are directly involved?

(A) Plotting the trajectory of the particle.

(B) Calculating the distance traveled by the particle (involving integrals, but based on coordinate path).

(C) Determining the velocity vector using derivatives of coordinate functions.

(D) All of the above.

Question 5. A navigation system uses GPS (Global Positioning System) data. How does coordinate geometry play a role in GPS?

(A) Representing the location of a point on Earth's surface using coordinates (latitude, longitude, altitude, which can be related to Cartesian). The system also uses distances calculated from satellite signals.

(B) Calculating the distance between the GPS receiver and multiple satellites based on signal travel time.

(C) Using trilateration (based on distances) to determine the receiver's position.

(D) All of the above.