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 coordinates of $P$ are

• A. $\left(2k,2k\right)$
• B. $\left(k,k\right)$
• C. $\left(a,2k\right)$
• D. $\left(k,2a\right)$

Answer 1

Answer: (A)

Parabolas $y^2=4a\left(x-k\right)$ and $x^2=4a\left(y-k\right)$ touch each other at the line $y=x$ ( $\because$ both parabolas are inverse of each other)
$\Rightarrow y=x$ is the common tangent at $P$
$\Rightarrow$ we get point $P$ by solving $y^2=4a\left(x-k\right)$ and $y=x$
$\Rightarrow$ point of contact $P=\left(2a,2a\right)$
$\Rightarrow {\left(2a\right)}^2=4a\left(2a-k\right)\Rightarrow k=a\Rightarrow P=\left(2k,2k\right)$