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Equation of Normal at Given Point

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

T^{2} \geq 8

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T^{2} \leq 6

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T \in (- \infty, - 8) \cup (8, \infty)

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T \in (- 8, 8)

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Solution

The correct option is A $T^{2} \geq 8$
Equation of normal to the parabola $y^{2} = 4 a x$ at the point $(a t^{2}, 2 a t)$ is
$y + t x = 2 a t + a t^{3} \dots (1)$
Equation $(1)$ cuts the parabola again at $(a T^{2}, 2 a T)$
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
$D \geq 0$
$\Rightarrow T^{2} - 4 \cdot 2 \cdot 1 \geq 0$
$\Rightarrow T^{2} \geq 8$

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