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Tangent and normal drawn to parabola at $\mathrm{A\left(a t^2, 2 a t\right)}$ , $\mathrm{t \neq 0}$ meet the x-axis at point $\mathrm{B}$ and $\mathrm{D}$ respectively. If the rectangle $\mathrm{A B C D}$ is completed, then locus of $\mathrm{C}$ is
Option: 1
$\mathrm{y=2a}$

Option: 2
$\mathrm{y+2a=0}$

Option: 3
$\mathrm{x=2a}$

Option: 4
$\mathrm{x+2a=0}$

Answers (1)

Equations of tangent and normal at $\mathrm{A}$ are $\mathrm{y t=x+a t^2 and y=-t x+2 a t+a t^3}$

$\mathrm{\Rightarrow \quad B \equiv\left(-a t^2, 0\right),D \equiv \left(2 a+a t^2, 0\right)}$

If $\mathrm{A B C D}$ is a rectangle, then mid-points of $\mathrm{ B D}$ and $\mathrm{ AC}$ will be coincident.

$\begin{array}{ll} \Rightarrow & \mathrm{h+a t^2=2 a+a t^2-a t^2, k+2 a t=0 }\\ \Rightarrow &\mathrm{ h=2 a, t=-\frac{k}{2 a}} \end{array}$
Thus locus is $\mathrm{x=2 a}$

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Sumit Saini

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