the locus of the mirror image of a point on the parabola
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 :

1. (1) $x-y=1$

2. (2) $2 x+y=5$

3. (3) $x+3 y=5$

4. (4) $x+2 y=4$

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 $(2,1)$

$\Rightarrow y-1=1(x-2)$

$\Rightarrow x-y=1$