Find the equation of normal to the parabola $y^{2} = 8 x at (8, 8)$ using parametric form.

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Solution

Given,
Parabola: $y^{2} = 8 x \Rightarrow a = 2$
Point on parabola $\equiv (8, 8)$
We know that
Parametric point $\equiv (a t^{2}, 2 a t)$
On camparing
$∴ a t^{2} = 8 and 2 a t = 8 \Rightarrow t = 2$

Equation of Normal to parabola at $(a t^{2}, 2 a t) is y = - t x + a t^{3} + 2 a t$

$\Rightarrow y = - (2) x + 2 \times 2^{3} + 2 \times 2 \times 2$

$\Rightarrow y = - 2 x + 16 + 8$

$\Rightarrow 2 x + y = 24$
Hence, the corrrect answer is Option a .

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