Question:
Let $\mathrm{C}$ be the locus of the mirror image of a point on the parabola $\mathrm{y}^{2}=4 \mathrm{x}$ with respect to the line $\mathrm{y}=\mathrm{x}$. Then the equation of tangent to $\mathrm{C}$ at $\mathrm{P}(2,1)$ is :
Correct Option: 1,
Solution:
Given $y^{2}=4 x$
Mirror image on $y=x \Rightarrow C: x^{2}=4 y$
$2 \mathrm{x}=4 \cdot \frac{\mathrm{dy}}{\mathrm{dx}} \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{x}}{2}$
$\left.\frac{\mathrm{dy}}{\mathrm{dx}}\right|_{\mathrm{P}(2,1)}=\frac{2}{2}=1$
Equation of tangent at $(2,1)$
$\Rightarrow y-1=1(x-2)$
$\Rightarrow x-y=1$