Question

If the axes are rotated through an angle of ${90}^{\circ }$ in any direction then the transformed equation of $x^2=4ay$ can be

• A. $Y^2=4aX$
• B. $Y^2=-4aX$
• C. $X^2=4aY$
• D. $X^2=-4aY$

Answer 1

Answer: (A), (B)

The correct options are
A $Y^2=4aX$
B $Y^2=-4aX$

Rotating the coordinate axes by ${90}^{\circ }$ anticlockwise will transform the coordinates as
$x=X\cos \left({90}^{\circ }\right)-Y\sin \left({90}^{\circ }\right)$ and
$y=X\sin \left({90}^{\circ }\right)+Y\cos \left({90}^{\circ }\right)$
$\Rightarrow x=-Y,y=X$

Substituting these values in the given equation
$(-Y{)}^2=4a(X)$
$\Rightarrow Y^2=4aX$
$\therefore$ the transformed equation is $Y^2=4aX$

Rotating the coordinate axes by ${90}^{\circ }$ clockwise will transform the coordinates as
$x=X\cos (-{90}^{\circ })-Y\sin (-{90}^{\circ })$ and
$y=X\sin (-{90}^{\circ })+Y\cos (-{90}^{\circ })$
$\Rightarrow x=Y,y=-X$

Substituting these values in the given equation
$(Y{)}^2=4a(-X)$
$\Rightarrow Y^2=-4aX$

$\therefore$ the transformed equation is $Y^2=-4aX$