Question:
Let $\mathrm{C}$ be the locus of the mirror image of a point on the parabola $y^{2}=4 x$ 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 $\mathrm{y}=\mathrm{x} \Rightarrow \mathrm{C}: \mathrm{x}^{2}=4 \mathrm{y}$
$2 x=4 \cdot \frac{d y}{d x} \Rightarrow \frac{d y}{d x}=\frac{x}{2}$
$\left.\frac{d y}{d x}\right|_{P(2,1)}=\frac{2}{2}=1$
Equation of tangent at $(2,1)$
$\Rightarrow \mathrm{y}-1=1(\mathrm{x}-2)$
$\Rightarrow x-y=1$
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