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Q.
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
NTA AbhyasNTA Abhyas 2020Conic Sections
Solution:
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 $P=\left(2 a , 2 a\right)$
$\Rightarrow \left(2 a\right)^{2}=4a\left(2 a - k\right)\Rightarrow k=a\Rightarrow P=\left(2 k , 2 k\right)$