What is Transformation of Axes?

What is Transformation of Axes?

(1) Shifting of origin without rotation of axes :

Let P ≡ (x, y) with respect to axes OX and OY.

Let O’ ≡ (α, β) with respect to axes OX and OY and let P ≡ (x’, y’) with respect to axes O’X’ and O’Y’, where OX and O’X’ are parallel and OY and O’Y’ are parallel.

Then x = x’ + α, y = y’ + β

or x’ = x – α, y’ = y – β

Thus if origin is shifted to point (α, β) without rotation of axes, then new equation of curve can be obtained by putting x + α in place of x and y + β in place of y.

(2) Rotation of axes without changing the origin :

Let O be the origin. Let P ≡ (x, y) with respect to axes OX and OY and let P ≡ (x’, y’) with respect to axes OX¢ and OY¢ where ∠X’OX = ∠Y’OY = θ.

then x = x’ cos θ – y’ sin θ

y = x’ sin θ + y’ cos θ

and x’ = x cos θ + y sin θ

y’ = –x sin θ + y cos θ

The above relation between (x, y) and (x’, y’) can be easily obtained with the help of following table

(3) Change of origin and rotation of axes :

If origin is changed to O’(α, β) and axes are rotated about the new origin O’ by an angle θ in the anti-clockwise sense such that the new co-ordinates of P(x, y) become (x’, y’) then the equations of transformation will be x = α  + x’ cos θ – y’ sin θ and y = β + x’ sin θ + y’ cos θ.

(4) Reflection (Image of a point) :

Let (x, y) be any point, then its image with respect to

1. x  axis ⇒ (x, –y)
2. y-axis ⇒ (–x, y)
3. origin ⇒ (–x, –y)
4. line y = x ⇒ (y, x)