Question

If two parabolas $y^{2}=4a\left(x - k\right)$ and $x^{2}=4a\left(y - k\right)$ have only one common point $P$ , then the equation of normal to $y^{2}=4a\left(x - k\right)$ at $P$ is

• A. $y+x=4a$
• B. $y+x=2a$
• C. $y+x=4$
• D. $y+x=2$

Answer 1

Answer: (A)

Parabolas $y^{2}=4a\left(x - k\right)$ and $x^{2}=4a\left(y - k\right)$ touche each other at line $y=x$
( $\because $ both parabola are inverse of each other)
$\Rightarrow $ $y=x$ is a common tangent at $P$
$\Rightarrow $ we get point $P$ by solving $y^{2}=4a\left(x - k\right)$ and $y=x$
Since, line touches parabola the quadratic obtained will have its discriminant as zero, hence $k=a$
$\Rightarrow $ point of contact $P$ is $\left(2 a , 2 a\right)$
$\Rightarrow $ equation of normal at $P\left(2 a , 2 a\right)$ is $x+y=4a$