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| Concept of Shifting the Origin | Formulae for Coordinate Transformation due to Shifting | Effect of Shifting Origin on Equations of Curves |
Transformation of Coordinates: Shifting of Origin
Concept of Shifting the Origin
In coordinate geometry, the position of a point or the equation of a curve is defined with respect to a chosen coordinate system, which includes an origin and a set of axes. Often, it is beneficial to change this coordinate system to simplify the representation of a geometric figure or to analyze its properties more easily. One fundamental type of coordinate transformation is the shifting of the origin, also known as translation of axes.
This transformation involves moving the origin of the coordinate system from its initial position, typically the point $(0, 0)$, to a new point $(h, k)$ with respect to the original axes. The crucial aspect of shifting the origin (translation) is that the orientation of the coordinate axes remains unchanged. The new x-axis is parallel to the old x-axis, and the new y-axis is parallel to the old y-axis.
Imagine you have a standard graph paper with its origin at the center. If you shift the origin, it's like picking up a second identical sheet of graph paper and placing its origin at a specific point $(h, k)$ on the first sheet, ensuring the grid lines on both sheets remain parallel. The points in the plane can now be described using coordinates from either grid.
Let the original coordinate system be denoted by $XOY$, with origin at $O(0, 0)$. Let the new coordinate system, after shifting the origin, be denoted by $X'O'Y'$, with the new origin at $O'(h, k)$ relative to the old system.
Any point $P$ in the plane can now be described by two sets of coordinates:
- $(x, y)$: These are the coordinates of point P with respect to the original axes $XOY$.
- $(X, Y)$: These are the coordinates of point P with respect to the new axes $X'O'Y'$.
The coordinate transformation formulae relate the old coordinates $(x, y)$ to the new coordinates $(X, Y)$ based on the location of the new origin $(h, k)$. This relationship allows us to convert the equation of a curve from one system to another, or to analyze points based on their position relative to the new origin.
Purpose of Shifting the Origin
The primary reasons for performing a translation of axes include:
- Simplifying Equations: Many equations of curves (like circles, parabolas, ellipses, hyperbolas) become significantly simpler when their center (for circle, ellipse, hyperbola) or vertex (for parabola) is located at the origin. Shifting the origin to such a key point can reduce the complexity of the equation, making analysis easier. For instance, the equation of a circle centered at $(h, k)$ is $(x-h)^2 + (y-k)^2 = r^2$, but in the new system with origin at $(h, k)$, its equation is simply $X^2 + Y^2 = r^2$.
- Choosing a Convenient Reference: In many problems, especially in physics and engineering, it is more convenient to study motion or forces relative to a specific point other than the standard origin. Shifting the origin allows us to set this convenient point as the new reference (new origin).
Understanding the concept of shifting the origin is a crucial first step before delving into the formulae that govern the transformation of coordinates.
Shifting the Origin (Concept Summary for Competitive Exams)
Definition:
A coordinate transformation where the origin is moved to a new point, but the direction of the coordinate axes remains unchanged (new axes are parallel to old axes).
Parameters:
Let the new origin be $O'(h, k)$ with respect to the original axes $XOY$.
Coordinates of a Point P:
- $(x, y)$: Old coordinates (w.r.t. $O$)
- $(X, Y)$: New coordinates (w.r.t. $O'$)
Purpose:
To simplify the equation of a curve or to choose a more convenient reference point for analysis.
Key Idea:
Translating the entire coordinate grid so that the point $(h, k)$ in the old system becomes the origin $(0, 0)$ in the new system, without rotating the grid.
Formulae for Coordinate Transformation due to Shifting
When the origin of the Cartesian coordinate system is shifted to a new location without changing the direction of the axes, the coordinates of any point in the plane change. It is essential to establish the relationship between the coordinates of a point in the original system and its coordinates in the new system. This relationship is given by the coordinate transformation formulae.
Let's define the two coordinate systems involved:
- The original coordinate system is $XOY$, with the origin at $O(0, 0)$.
- The new coordinate system is $X'O'Y'$, obtained by shifting the origin to a new point $O'(h, k)$ with respect to the original system $XOY$. The $X'$-axis is parallel to the $X$-axis, and the $Y'$-axis is parallel to the $Y$-axis.
Consider an arbitrary point P in the plane. Let its coordinates be:
- $(x, y)$ with respect to the original system $XOY$. These are the old coordinates.
- $(X, Y)$ with respect to the new system $X'O'Y'$. These are the new coordinates.
Derivation of the Transformation Formulae
We can derive the relationship between $(x, y)$, $(X, Y)$, and $(h, k)$ by considering the position of point P relative to both origins and sets of axes.
Refer to the diagram illustrating the shift of the origin.
Draw a perpendicular line from point P to the original x-axis, meeting it at point M. The distance OM represents the x-coordinate of P in the old system, so $OM = x$.
Draw a perpendicular line from point P to the original y-axis, meeting it at point N. The distance ON represents the y-coordinate of P in the old system, so $ON = y$.
Similarly, draw a perpendicular line from point P to the new x'-axis, meeting it at point M'. The distance O'M' represents the x-coordinate of P in the new system, so $O'M' = X$.
Draw a perpendicular line from point P to the new y'-axis, meeting it at point N'. The distance O'N' represents the y-coordinate of P in the new system, so $O'N' = Y$.
The coordinates of the new origin $O'$ with respect to the old origin $O$ are given as $(h, k)$. This means the horizontal distance of $O'$ from the old y-axis is $h$, and the vertical distance of $O'$ from the old x-axis is $k$.
Now, let's analyze the distances along the axes:
The total horizontal distance from the old origin O to the point M (which is the x-coordinate of P in the old system) can be seen as the sum of the horizontal distance from O to the new origin O' (which is $h$) and the horizontal distance from the new origin O' to the point M' (which is the x-coordinate of P in the new system, $X$).
Thus, we have the relationship:
$OM = OO' \text{ projected on X-axis } + O'M'$
(Horizontal distances)
$x = h + X$
... (i)
Similarly, the total vertical distance from the old origin O to the point N (which is the y-coordinate of P in the old system) can be seen as the sum of the vertical distance from O to the new origin O' (which is $k$) and the vertical distance from the new origin O' to the point N' (which is the y-coordinate of P in the new system, $Y$).
Thus, we have the relationship:
$ON = OO' \text{ projected on Y-axis } + O'N'$
(Vertical distances)
$y = k + Y$
... (ii)
Equations (i) and (ii) are the fundamental coordinate transformation formulae when the origin is shifted to $(h, k)$ without rotation of axes.
Transformation Formulae Summary
If the origin is shifted to the point $(h, k)$ relative to the original axes $XOY$, and the directions of the axes remain unchanged, then the coordinates of a point P change from $(x, y)$ in the original system to $(X, Y)$ in the new system. The relationship between these coordinates is given by:
To find the old coordinates $(x, y)$ from the new coordinates $(X, Y)$:
$\mathbf{x = X + h}$
$\mathbf{y = Y + k}$
To find the new coordinates $(X, Y)$ from the old coordinates $(x, y)$:
These formulae are simply rearrangements of the above equations:
$X = x - h$
... (iii)
$Y = y - k$
... (iv)
These formulae are crucial for problems involving the transformation of coordinates due to translation of the origin.
Example 1. If the origin is shifted to the point (2, -3), what are the new coordinates of the point P(5, 1) in the original system?
Answer:
Given:
Old coordinates of the point P: $(x, y) = (5, 1)$.
Coordinates of the new origin: $(h, k) = (2, -3)$. Here, $h=2$ and $k=-3$.
We need to find the new coordinates $(X, Y)$ of point P.
Using the transformation formulae for finding new coordinates from old coordinates:
$X = x - h$
... (i)
$Y = y - k$
... (ii)
Substitute the given values into equations (i) and (ii):
$X = 5 - 2 = 3$
$Y = 1 - (-3) = 1 + 3 = 4$
Therefore, the new coordinates of the point P are $\mathbf{(3, 4)}$.
Example 2. If the origin is shifted to (-1, 4), find the original coordinates of the point Q whose new coordinates are (6, -2).
Answer:
Given:
New coordinates of the point Q: $(X, Y) = (6, -2)$. Here, $X=6$ and $Y=-2$.
Coordinates of the new origin with respect to the old system: $(h, k) = (-1, 4)$. Here, $h=-1$ and $k=4$.
We need to find the original coordinates $(x, y)$ of point Q.
Using the transformation formulae for finding old coordinates from new coordinates:
$x = X + h$
... (i)
$y = Y + k$
... (ii)
Substitute the given values into equations (i) and (ii):
$x = 6 + (-1) = 6 - 1 = 5$
$y = -2 + 4 = 2$
Therefore, the original coordinates of the point Q are $\mathbf{(5, 2)}$.
Coordinate Transformation Formulae (Shifting Origin) Summary
Scenario:
Origin shifts from $O(0,0)$ to $O'(h, k)$ in the old system. Axes remain parallel.
Notation:
Point P has old coordinates $(x, y)$ and new coordinates $(X, Y)$.
Formulae:
- To find old coordinates from new:
$x = X + h$
$y = Y + k$
- To find new coordinates from old:
$X = x - h$
$Y = y - k$
Key Idea:
An old coordinate is the new coordinate plus the coordinate of the new origin in the old system (or vice versa, new is old minus the shift).
Effect of Shifting Origin on Equations of Curves
When the coordinate system is translated by shifting the origin to a new point $(h, k)$, the coordinates of every point in the plane change. This change in coordinates necessitates a corresponding change in the algebraic equation that represents any given curve.
Suppose we have a curve represented by the equation $f(x, y) = 0$ in the original coordinate system $XOY$. If the origin is shifted to $O'(h, k)$, and the new coordinates of a point P are $(X, Y)$, while its old coordinates are $(x, y)$, we know the relationship:
$x = X + h$
(Transformation formula)
$y = Y + k$
(Transformation formula)
To find the equation of the curve in the new coordinate system $X'O'Y'$, we substitute these expressions for $x$ and $y$ into the original equation $f(x, y) = 0$.
$f(X+h, Y+k) = 0$
... (i)
Equation (i) is the equation of the same curve, but now expressed in terms of the new coordinates $(X, Y)$. It is a standard convention to use lowercase letters $x$ and $y$ to denote the coordinates in the current system being used. Therefore, after obtaining the equation in terms of $X$ and $Y$, we usually replace $X$ with $x$ and $Y$ with $y$ to represent the transformed equation as $f(x+h, y+k) = 0$, where $(x, y)$ now refer to the coordinates in the new system.
Conversely, if you are given the equation of a curve in the new system, say $F(X, Y) = 0$, and you want to find its equation in the original system, you would use the inverse transformation formulae: $X = x - h$ and $Y = y - k$. Substituting these into the new equation gives $F(x-h, y-k) = 0$, which is the equation in the original system $(x, y)$.
The process of shifting the origin is a powerful technique, particularly useful for simplifying equations of curves. By choosing the new origin $(h, k)$ strategically (e.g., at the center or vertex of the curve), we can often eliminate linear terms, making the equation easier to recognize and analyze.
Example 1. Find the new equation of the line $2x - 3y + 5 = 0$ when the origin is shifted to the point (1, -2).
Answer:
Given:
The original equation of the line is $2x - 3y + 5 = 0$.
The new origin is the point $(h, k) = (1, -2)$. So, $h=1$ and $k=-2$.
Let $(x, y)$ be the coordinates of a point on the line in the original system, and let $(X, Y)$ be the coordinates of the same point in the new system.
The transformation formulae for shifting the origin are:
$x = X + h$
... (i)
$y = Y + k$
... (ii)
Substitute the values of $h$ and $k$ into equations (i) and (ii):
$x = X + 1$
... (iii)
$y = Y - 2$
... (iv)
Now, substitute these expressions for $x$ and $y$ from equations (iii) and (iv) into the original equation of the line $2x - 3y + 5 = 0$:
$2(X + 1) - 3(Y - 2) + 5 = 0$
... (v)
Simplify equation (v) by expanding and combining terms:
$2X + 2 - 3Y + 6 + 5 = 0$
$2X - 3Y + (2 + 6 + 5) = 0$
$2X - 3Y + 13 = 0$
... (vi)
Equation (vi) is the equation of the line in the new coordinate system $(X, Y)$. By convention, we rewrite the equation using lowercase $x$ and $y$ to represent coordinates in the new system.
The new equation of the line is $\mathbf{2x - 3y + 13 = 0}$.
Note on Slope:
The slope of the original line $2x - 3y + 5 = 0$ is $m = -\frac{2}{-3} = \frac{2}{3}$. The slope of the new equation $2x - 3y + 13 = 0$ is also $m = -\frac{2}{-3} = \frac{2}{3}$. This is expected because shifting the origin only translates the coordinate grid; it does not rotate it, so the orientation and slope of the line remain unchanged relative to the axes.
Example 2. Find the point to which the origin should be shifted so that the equation $x^2 + y^2 - 6x + 8y + 21 = 0$ is transformed into an equation with no first-degree terms (i.e., terms containing $x$ or $y$). Also, find the transformed equation.
Answer:
Given the original equation: $x^2 + y^2 - 6x + 8y + 21 = 0$.
Let the origin be shifted to the point $(h, k)$. The transformation formulae are $x = X + h$ and $y = Y + k$, where $(X, Y)$ are the coordinates in the new system.
Substitute these expressions for $x$ and $y$ into the given equation:
$(X + h)^2 + (Y + k)^2 - 6(X + h) + 8(Y + k) + 21 = 0$
... (i)
Expand the terms in equation (i):
$(X^2 + 2hX + h^2) + (Y^2 + 2kY + k^2) - (6X + 6h) + (8Y + 8k) + 21 = 0$
Rearrange and group terms by powers of X and Y:
$X^2 + Y^2 + (2h - 6)X + (2k + 8)Y + (h^2 + k^2 - 6h + 8k + 21) = 0$
... (ii)
For the transformed equation (ii) to have no first-degree terms (terms containing $X$ or $Y$), the coefficients of $X$ and $Y$ must be zero.
Set the coefficient of X to zero:
$2h - 6 = 0$
... (iii)
Solve equation (iii) for $h$:
$2h = 6 \implies h = 3$
$h = 3$
... (iv)
Set the coefficient of Y to zero:
$2k + 8 = 0$
... (v)
Solve equation (v) for $k$:
$2k = -8 \implies k = -4$
$k = -4$
... (vi)
From (iv) and (vi), the coordinates of the point to which the origin should be shifted are $(h, k) = (3, -4)$.
Therefore, the origin should be shifted to the point $\mathbf{(3, -4)}$.
Now, we find the transformed equation by substituting $h=3$ and $k=-4$ back into equation (ii). Since the coefficients of $X$ and $Y$ are zero, the equation simplifies to:
$X^2 + Y^2 + (\text{Constant term}) = 0$
Calculate the value of the constant term: $h^2 + k^2 - 6h + 8k + 21$
Constant term $= (3)^2 + (-4)^2 - 6(3) + 8(-4) + 21$
... (vii)
Simplify equation (vii):
Constant term $= 9 + 16 - 18 - 32 + 21$
Constant term $= 25 - 18 - 32 + 21$
Constant term $= 7 - 32 + 21$
Constant term $= -25 + 21$
Constant term $= -4$
... (viii)
Substitute the constant term value from equation (viii) into the simplified equation $X^2 + Y^2 + (\text{Constant term}) = 0$:
$X^2 + Y^2 - 4 = 0$
... (ix)
Equation (ix) is the transformed equation in the new coordinate system $(X, Y)$. Replacing $(X, Y)$ with $(x, y)$ for standard notation:
The transformed equation is $\mathbf{x^2 + y^2 = 4}$.
Interpretation:
The original equation $x^2 + y^2 - 6x + 8y + 21 = 0$ represents a circle. By completing the square, we can rewrite it as $(x^2 - 6x + 9) + (y^2 + 8y + 16) + 21 - 9 - 16 = 0$, which simplifies to $(x - 3)^2 + (y + 4)^2 = 4$. This is the equation of a circle with center $(3, -4)$ and radius $2$. When the origin is shifted to the center $(3, -4)$, the equation in the new system becomes $X^2 + Y^2 = 2^2 = 4$. This confirms our result and the geometric meaning of eliminating the first-degree terms.
Alternative Method (Completing the Square):
The coordinates $(h, k)$ to which the origin must be shifted to eliminate first-degree terms in a quadratic equation of the form $Ax^2+By^2+Dx+Ey+F=0$ (where $A, B \neq 0$) are often found by simply completing the square for the $x$ and $y$ terms.
Given equation: $x^2 + y^2 - 6x + 8y + 21 = 0$.
Group the terms involving $x$ and terms involving $y$:
$(x^2 - 6x) + (y^2 + 8y) + 21 = 0$
... (x)
Complete the square for each group. For $x^2 - 6x$, add $(\frac{-6}{2})^2 = (-3)^2 = 9$. For $y^2 + 8y$, add $(\frac{8}{2})^2 = (4)^2 = 16$. Add and subtract these values to maintain equality in equation (x):
$(x^2 - 6x + 9) + (y^2 + 8y + 16) + 21 - 9 - 16 = 0$
... (xi)
Rewrite the perfect square trinomials as squares and simplify the constants in equation (xi):
$(x - 3)^2 + (y + 4)^2 + 21 - 25 = 0$
... (xii)
$(x - 3)^2 + (y + 4)^2 - 4 = 0$
... (xiii)
$(x - 3)^2 + (y - (-4))^2 = 4$
... (xiv)
Equation (xiv) is in the form $(x-h)^2 + (y-k)^2 = r^2$, where $(h, k)$ is the center of the circle. Comparing equation (xiv) with this standard form, we see that $h=3$ and $k=-4$. Thus, the center of the original equation is $(3, -4)$.
Shifting the origin to $(h, k) = (3, -4)$ means the new coordinates $(X, Y)$ are related to the old coordinates $(x, y)$ by $X = x - 3$ and $Y = y - (-4) = y + 4$.
Substitute $X = x - 3$ and $Y = y + 4$ into equation (xiv):
$X^2 + Y^2 = 4$
... (xv)
Replacing $(X, Y)$ with $(x, y)$ for standard notation in the new system, the transformed equation is $\mathbf{x^2 + y^2 = 4}$. This confirms the previous result.
Effect of Shifting Origin on Equations (Summary)
Transformation Rule:
If origin is shifted to $(h, k)$, replace $x$ with $X+h$ and $y$ with $Y+k$ in the original equation $f(x, y) = 0$. The new equation is $f(X+h, Y+k)=0$. (Replace $(X, Y)$ with $(x, y)$ in the final form).
Inverse Transformation:
If equation in new system $(X, Y)$ is $F(X, Y) = 0$, replace $X$ with $x-h$ and $Y$ with $y-k$ to get the original equation $F(x-h, y-k)=0$.
Purpose:
Simplify equations, especially conic sections, by moving their center or vertex to the new origin. This eliminates first-degree terms in equations of circles, ellipses, and hyperbolas when shifting to the center, and one linear term in a parabola when shifting to the vertex.
Eliminating First-Degree Terms:
For a general quadratic equation $Ax^2 + By^2 + Dx + Ey + F = 0$ (where A and B are not both zero), shifting the origin to $(h, k)$ eliminates the linear terms if:
- $2Ah + D = 0 \implies h = -D/(2A)$
- $2Bk + E = 0 \implies k = -E/(2B)$
This point $(-D/(2A), -E/(2B))$ is the center for central conics (Ellipse, Hyperbola) or can be found via completing the square.