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Consider a pa...

Question

Consider a parabola $y^{2} = 4 a x .$ If the normal to the parabola at the point $(a t^{2}, 2 a t)$ cuts the parabola again at $(a T^{2}, 2 a T)$ , then

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Solution

Equation of normal to the parabola $y^{2} = 4 a x$ at the point $(a t^{2}, 2 a t)$ is
$y + t x = 2 a t + a t^{3} \dots (i)$
Equation $(i)$ cuts the parabola again $(a T^{2}, 2 a T) .$
Then, $2 a T + t a T^{2} = 2 a t + a t^{3}$
$\Rightarrow 2 a (T - t) = - a t (T^{2} - t^{2})$
$\Rightarrow 2 = - t (T + t) (∵ t \neq T)$
$\Rightarrow t^{2} + t T + 2 = 0$
Since $t$ is real
$∴ D \geq 0$
$\Rightarrow T^{2} - 4 \cdot 2 \cdot 1 \geq 0$
$∴ T^{2} \geq 8$

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