First slide

Parabola

Question

Let $Q$ be the foot of the perpendicular from the
origin $O$ to the tangent at a point $p (α, β)$ on the
parabola $y^{2} = 4 a x,$ and $S$ be the focus of the parabola,
then $(O Q)^{2} (S P)$ is equal to

Moderate

A

$α$
B

$a α^{2}$
C

$β$
D

$a β^{2}$

Solution

Let $α = a t^{2}, β = 2 a t$
Equation of the tangent at $P (α, β)$ is $t y = x + a t^{2}$
$\begin{array}{l} \Rightarrow O Q = \frac{a t^{2}}{\sqrt{1 + t^{2}}} \\ S P = a (t^{2} + 1) \\ So (O Q)^{2} (S P) = \frac{a^{2} t^{4}}{1 + t^{2}} \times a (1 + t^{2}) = a^{3} t^{4} = a α^{2} \end{array}$

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