Circle in Coordinate Geometry

Equation of a Circle: Standard Form $(x-h)^2 + (y-k)^2 = r^2$

The circle is a fundamental conic section and one of the simplest curves to describe geometrically and algebraically. In coordinate geometry, the equation of a circle directly translates its geometric definition into an algebraic relationship between the coordinates of points lying on the circle.

Definition of a Circle

A circle is defined as the locus of all points in a plane that are at a constant distance from a fixed point in the same plane. The fixed point is called the center of the circle, and the constant distance is called the radius of the circle. The radius must be a positive real number ($r > 0$).

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Derivation of the Standard Form (Center-Radius Form)

Let $C$ be the center of the circle with coordinates $(h, k)$.

Let $r$ be the radius of the circle, where $r > 0$.

Let $P$ be any arbitrary point on the circumference of the circle, with coordinates $(x, y)$.

According to the definition of a circle, the distance between the center $C(h, k)$ and any point $P(x, y)$ on the circle is always equal to the radius $r$.

We can express the distance between points $C$ and $P$ using the distance formula in two dimensions:

By the definition of the circle, this distance is equal to the radius $r$. So, we set $|CP| = r$:

To eliminate the square root and get a simpler algebraic equation, we square both sides of equation (i):

Equation (ii) is an equation that is satisfied by the coordinates $(x, y)$ of every point $P$ that lies on the circle with center $(h, k)$ and radius $r$. Furthermore, any point $(x, y)$ that satisfies this equation must be at a distance $r$ from $(h, k)$ (taking the positive square root), and thus lies on the circle. Therefore, equation (ii) is the equation of the circle.

Standard Form (Center-Radius Form)

The equation $(x - h)^2 + (y - k)^2 = r^2$ is called the standard form or center-radius form of the equation of a circle. It is particularly useful because the coordinates of the center $(h, k)$ and the value of the radius $r$ can be read directly from the equation.

$\mathbf{(x - h)^2 + (y - k)^2 = r^2}$

Special Case: Circle centered at the Origin

If the center of the circle is located at the origin $O(0, 0)$, then the coordinates of the center are $h=0$ and $k=0$. Substituting these values into the standard form:

$(x - 0)^2 + (y - 0)^2 = r^2$

This is the simplest form of the equation of a circle, representing a circle centered at the origin with radius $r$.

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Answer:

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General Equation of a Circle $x^2 + y^2 + 2gx + 2fy + c = 0$

The standard form of the equation of a circle, $(x - h)^2 + (y - k)^2 = r^2$, is useful for easily identifying the center and radius. By expanding this equation, we can obtain another important form called the general equation of a circle.

Derivation of the General Form

Start with the standard form of a circle with center $(h, k)$ and radius $r$:

Expand the squared terms in equation (i) using the formula $(a-b)^2 = a^2 - 2ab + b^2$:

Rearrange the terms in equation (ii) to group $x^2$ and $y^2$ terms first, followed by terms with $x$ and $y$, and finally the constant terms:

Equation (iii) is a second-degree equation in $x$ and $y$. To express it in a more standard and compact general form, we introduce new constants based on the coefficients:

• Let the coefficient of $x$ be $2g$. So, $2g = -2h \implies h = -g$.
• Let the coefficient of $y$ be $2f$. So, $2f = -2k \implies k = -f$.
• Let the constant term be $c$. So, $c = h^2 + k^2 - r^2$.

From the substitution for $c$, we can also express the radius squared $r^2$ in terms of $g$, $f$, and $c$:

$c = (-g)^2 + (-f)^2 - r^2 \implies c = g^2 + f^2 - r^2$

Substitute the new constants $2g$, $2f$, and $c$ into equation (iii):

$\mathbf{x^2 + y^2 + 2gx + 2fy + c = 0}$

This is the general equation of a circle. It is a second-degree equation in $x$ and $y$ with specific characteristics: the coefficients of $x^2$ and $y^2$ are equal (and typically taken as 1 by dividing the entire equation if they are non-zero and equal), and there is no $xy$ term.

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Finding Center and Radius from General Form

Given the general equation of a circle $x^2 + y^2 + 2gx + 2fy + c = 0$, we can determine the coordinates of its center and the length of its radius. There are two common methods:

Method 1: Comparing Coefficients

This method uses the relationships established during the derivation:

• The x-coordinate of the center is $h = -g$. Since $2g$ is the coefficient of the $x$ term in the general equation, $g$ is half of the coefficient of $x$. So, $h = -(\frac{\text{Coefficient of } x}{2})$.
• The y-coordinate of the center is $k = -f$. Since $2f$ is the coefficient of the $y$ term, $f$ is half of the coefficient of $y$. So, $k = -(\frac{\text{Coefficient of } y}{2})$.
• The radius squared is $r^2 = g^2 + f^2 - c$. The radius is the square root of this value.

So, the center of the circle is $\mathbf{(-g, -f)}$ and the radius is $\mathbf{r = \sqrt{g^2 + f^2 - c}}$.

Method 2: Completing the Square

This method transforms the general equation back into the standard form $(x - h)^2 + (y - k)^2 = r^2$ by completing the square for the terms involving $x$ and the terms involving $y$.

Start with the general equation: $x^2 + y^2 + 2gx + 2fy + c = 0$.

Group the terms involving $x$ together and the terms involving $y$ together, and move the constant term to the right side:

Complete the square for the $x$ terms: Take half of the coefficient of $x$ ($2g/2 = g$) and square it ($g^2$). Add $g^2$ to both sides of the equation.

Complete the square for the $y$ terms: Take half of the coefficient of $y$ ($2f/2 = f$) and square it ($f^2$). Add $f^2$ to both sides of the equation.

Rewrite the perfect square trinomials as squares:

Equation (vii) is now in the standard form $(x - h)^2 + (y - k)^2 = r^2$.

Comparing equation (vii) with the standard form, we have:

• $x - h = x + g \implies h = -g$
• $y - k = y + f \implies k = -f$
• $r^2 = g^2 + f^2 - c \implies r = \sqrt{g^2 + f^2 - c}$

Both methods yield the same result for the center and radius.

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Conditions for a Real Circle

The expression for the radius $r = \sqrt{g^2 + f^2 - c}$ imposes a condition for the equation $x^2 + y^2 + 2gx + 2fy + c = 0$ to represent a real circle with a positive radius ($r > 0$). The value under the square root must be positive.

• If $g^2 + f^2 - c > 0$: The radius $r = \sqrt{g^2 + f^2 - c}$ is a positive real number. The equation represents a real circle.
• If $g^2 + f^2 - c = 0$: The radius $r = \sqrt{0} = 0$. The equation $(x+g)^2 + (y+f)^2 = 0$ is only satisfied by the single point $(x, y) = (-g, -f)$. This is called a point circle or a degenerate circle.
• If $g^2 + f^2 - c < 0$: The value under the square root is negative, so the radius $\sqrt{g^2 + f^2 - c}$ is an imaginary number. The equation $(x+g)^2 + (y+f)^2 = \text{negative value}$ has no real solutions for $(x, y)$. This represents an imaginary circle, which has no locus in the real Cartesian plane.

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Identifying the General Equation of a Circle

A general second-degree equation in $x$ and $y$ has the form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. For such an equation to represent a circle, it must satisfy specific conditions:

1. The coefficient of the $xy$ term must be zero ($B=0$). This means there are no terms like $xy$ in the equation.
2. The coefficients of the $x^2$ term and the $y^2$ term must be equal and non-zero ($A=C \neq 0$).

If these two conditions are met, the equation $Ax^2 + Ay^2 + Dx + Ey + F = 0$ can be converted into the standard general form $x^2 + y^2 + 2gx + 2fy + c = 0$ by simply dividing the entire equation by the common non-zero coefficient $A$.

$\frac{Ax^2}{A} + \frac{Ay^2}{A} + \frac{Dx}{A} + \frac{Ey}{A} + \frac{F}{A} = 0 \implies x^2 + y^2 + \left(\frac{D}{A}\right)x + \left(\frac{E}{A}\right)y + \left(\frac{F}{A}\right) = 0$

Comparing this with $x^2 + y^2 + 2gx + 2fy + c = 0$, we have $2g = D/A$, $2f = E/A$, and $c = F/A$. The center is $(-D/(2A), -E/(2A))$ and radius is $\sqrt{(D/(2A))^2 + (E/(2A))^2 - F/A}$.

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Answer:

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Equation of Circle in Various Forms (e.g., Diameter Form)

Besides the standard (center-radius) form and the general form, the equation of a circle can be expressed in other forms that are useful when different information about the circle is given. These forms are often derived from the basic geometric properties of a circle and the standard formulas of coordinate geometry.

1. Diameter Form

If the coordinates of the endpoints of a diameter of a circle are known, we can directly write the equation of the circle. Let the endpoints of a diameter of a circle be $A(x_1, y_1)$ and $B(x_2, y_2)$.

Consider any point $P(x, y)$ on the circumference of the circle, distinct from A and B. A fundamental geometric property of a circle is that the angle inscribed in a semicircle is always a right angle ($90^\circ$). Since AB is a diameter, the angle $\angle APB$ formed by connecting any point P on the circumference to A and B is $90^\circ$.

The segments AP and BP are perpendicular to each other (unless P coincides with A or B). For non-vertical and non-horizontal lines, this perpendicularity means the product of their slopes is -1.

Slope of AP ($m_{AP}$) = $\frac{y - y_1}{x - x_1}$ (if $x \neq x_1$)

Slope of BP ($m_{BP}$) = $\frac{y - y_2}{x - x_2}$ (if $x \neq x_2$)

Using the condition for perpendicular lines $m_{AP} \times m_{BP} = -1$:

Multiply both sides by $(x - x_1)(x - x_2)$ (assuming $x \neq x_1$ and $x \neq x_2$):

Rearrange equation (i) by moving the term from the right side to the left side:

This equation holds true for any point $P(x, y)$ on the circle. It also includes the cases where $x = x_1$ or $x = x_2$ (vertical segments), or $y = y_1$ or $y = y_2$ (horizontal segments). For example, if $x=x_1$, the first term is 0, and the equation becomes $(y - y_1)(y - y_2) = 0$, which means $y=y_1$ or $y=y_2$. The point is $(x_1, y_1)$ or $(x_1, y_2)$, which are points on the circle. Also, if $P$ coincides with A or B, the equation yields $0=0$.

Diameter Form Formula:

The equation of the circle with endpoints of a diameter $A(x_1, y_1)$ and $B(x_2, y_2)$ is:

$\mathbf{(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0}$

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Answer:

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2. Circle Passing Through Three Non-Collinear Points

Given three distinct non-collinear points in a plane, there exists a unique circle that passes through all three points. These points form a triangle, and the circle passing through them is the circumcircle of that triangle. Its center is the circumcenter, and its radius is the circumradius.

To find the equation of the circle passing through three non-collinear points $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$:

1. Assume the General Equation: Start with the general equation of a circle: $x^2 + y^2 + 2gx + 2fy + c = 0$. This equation involves three unknown coefficients: $g, f,$ and $c$.

2. Formulate a System of Equations: Since the points A, B, and C lie on the circle, their coordinates must satisfy the general equation. Substitute the coordinates of each of the three points into the general equation. This will yield three distinct linear equations in terms of $g, f,$ and $c$:

   For point A$(x_1, y_1)$: $x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c = 0$

   For point B$(x_2, y_2)$: $x_2^2 + y_2^2 + 2gx_2 + 2fy_2 + c = 0$

   For point C$(x_3, y_3)$: $x_3^2 + y_3^2 + 2gx_3 + 2fy_3 + c = 0$

3. Solve the System: Solve this system of three linear equations for the unknowns $g, f,$ and $c$. Standard methods like elimination or substitution can be used.

4. Substitute Back: Substitute the calculated values of $g, f,$ and $c$ back into the general equation $x^2 + y^2 + 2gx + 2fy + c = 0$. This gives the equation of the unique circle passing through the three points.

Alternative Method (Using Geometric Properties):

Another approach involves finding the circumcenter (center) and circumradius (radius) directly from the points:

1. Find the equations of the perpendicular bisectors of any two sides of the triangle (e.g., AB and BC).
2. The intersection point of these two perpendicular bisectors is the circumcenter $(h, k)$ of the triangle, which is the center of the circle.
3. Calculate the radius $r$ as the distance from the circumcenter $(h, k)$ to any one of the three given points (A, B, or C).
4. Substitute the coordinates of the center $(h, k)$ and the radius $r$ into the standard form $(x - h)^2 + (y - k)^2 = r^2$ to get the equation of the circle. This can then be expanded to the general form if needed. This method is conceptually clear but can be computationally intensive.

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3. Parametric Form

The equation of a circle can also be expressed using a parameter, typically an angle. This form is useful in calculus and other applications where representing coordinates as functions of a single variable is convenient.

For a circle with center $(h, k)$ and radius $r$, the parametric equations are derived using trigonometry. Consider a point $P(x, y)$ on the circle. Draw a line segment from the center $C(h, k)$ to $P(x, y)$. Let $\theta$ be the angle this segment makes with a line through $C$ parallel to the positive x-axis, measured counterclockwise.

The horizontal displacement from the center is $x - h$, and the vertical displacement is $y - k$. From trigonometry in the right triangle formed:

$x - h = r \cos \theta \implies \mathbf{x = h + r \cos \theta}$

$y - k = r \sin \theta \implies \mathbf{y = k + r \sin \theta}$

where the parameter $\theta$ usually ranges from $0$ to $2\pi$ (or $0^\circ$ to $360^\circ$) to cover the entire circle.

These equations $(x = h + r \cos \theta, y = k + r \sin \theta)$ are the parametric equations of the circle. For a circle centered at the origin $(0, 0)$, the parametric equations simplify to $x = r \cos \theta$ and $y = r \sin \theta$.

Note that $(x - h)^2 + (y - k)^2 = (r \cos \theta)^2 + (r \sin \theta)^2 = r^2 (\cos^2 \theta + \sin^2 \theta) = r^2 (1) = r^2$, confirming that the parametric form satisfies the standard equation.

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Geometrical Condition for the Intersection of a Line and a Circle

When considering a straight line and a circle in the same plane, there are only three possible ways they can be positioned relative to each other:

1. The line might cross through the circle, intersecting it at two distinct points. Such a line is called a secant line.

2. The line might just touch the circle at exactly one point. Such a line is called a tangent line.

3. The line might not touch or cross the circle at all. It lies entirely outside the circle.

These geometric relationships can be determined by comparing the radius of the circle with the distance from the center of the circle to the line. The distance from the center to the line is always the perpendicular distance.

Conditions Based on Distance from Center to Line

Let the circle be defined by its center $C(h, k)$ and radius $r$ (where $r > 0$).

Let the straight line be given by its general equation $L: Ax + By + C_{\text{line}} = 0$ (where $A^2 + B^2 \neq 0$).

Calculate the perpendicular distance, $d$, from the center of the circle $C(h, k)$ to the line $L$ using the formula for the distance of a point from a line:

Now, we compare this distance $d$ with the radius $r$ of the circle:

Condition	Geometric Situation	Number of Intersection Points
$d < r$	The perpendicular distance from the center to the line is less than the radius. The line is a secant and intersects the circle at two distinct points.	2
$d = r$	The perpendicular distance from the center to the line is exactly equal to the radius. The line is a tangent and touches the circle at exactly one point.	1
$d > r$	The perpendicular distance from the center to the line is greater than the radius. The line lies outside the circle and does not intersect it.	0

This geometric condition provides a simple and intuitive way to determine the relationship between a line and a circle without solving their equations algebraically (which would involve finding the roots of a quadratic equation).

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Answer:

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Relative Position of Two Circles

Given two circles in the same plane, they can be positioned relative to each other in several distinct ways. These positions determine whether the circles intersect, touch, or are entirely separate. The relationship between the distance between their centers and the sum/difference of their radii provides a definitive way to classify their relative position.

Setup

Let the two circles be:

• Circle 1, denoted by $C_1$, with center coordinates $(h_1, k_1)$ and radius $r_1$.
• Circle 2, denoted by $C_2$, with center coordinates $(h_2, k_2)$ and radius $r_2$.

Assume that both circles are real, meaning $r_1 > 0$ and $r_2 > 0$.

The key quantity to calculate is the distance $d$ between the centers of the two circles. We use the distance formula in two dimensions for points $C_1(h_1, k_1)$ and $C_2(h_2, k_2)$:

$\mathbf{d = |C_1 C_2| = \sqrt{(h_2 - h_1)^2 + (k_2 - k_1)^2}}$

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Conditions for Relative Positions

We compare the distance $d$ between the centers with the sum of the radii ($r_1 + r_2$) and the absolute difference of the radii ($|r_1 - r_2|$).

1. Circles are Separate (Outside each other): The distance between the centers is greater than the sum of their radii. This means there is a gap between the circles.

   • Condition: $\mathbf{d > r_1 + r_2}$
   • Number of Common Points (Intersection Points): 0
   

2. Circles Touch Externally: The distance between the centers is exactly equal to the sum of their radii. The circles meet at a single point on the line segment connecting their centers.

   • Condition: $\mathbf{d = r_1 + r_2}$
   • Number of Common Points: 1 (point of tangency)
   

3. Circles Intersect at Two Distinct Points: The distance between the centers is less than the sum of the radii, allowing overlap, but greater than the absolute difference of the radii (which prevents one circle from being completely inside the other). The circles cross each other at two distinct points.

   • Condition: $\mathbf{|r_1 - r_2| < d < r_1 + r_2}$
   • Number of Common Points: 2
   

4. Circles Touch Internally: The distance between the centers is exactly equal to the absolute difference of their radii. This assumes the circles are of different sizes ($r_1 \neq r_2$). The smaller circle touches the larger circle at a single point from the inside.

   • Condition: $\mathbf{d = |r_1 - r_2|}$ and $r_1 \neq r_2$
   • Number of Common Points: 1 (point of tangency)
   

5. One Circle Inside the Other (Without Touching): The distance between the centers is less than the absolute difference of their radii. This implies the smaller circle is strictly contained within the larger circle without touching its boundary. This also assumes the circles are of different sizes ($r_1 \neq r_2$).

   • Condition: $\mathbf{d < |r_1 - r_2|}$ and $r_1 \neq r_2$
   • Number of Common Points: 0
   

6. Concentric Circles: The centers of the two circles coincide, but their radii are different. One circle is strictly inside the other.

   • Condition: $\mathbf{d = 0}$ and $r_1 \neq r_2$
   • Number of Common Points: 0

   Note that this is a special case of condition 5 where $d=0$, since $0 < |r_1 - r_2|$ when $r_1 \neq r_2$.

7. Coincident Circles: The centers coincide, and the radii are equal. The two circles are identical.

   • Condition: $\mathbf{d = 0}$ and $r_1 = r_2$
   • Number of Common Points: Infinite

   This corresponds to both equations representing the same locus of points.

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Answer:

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