First slide

Parabola

Question

Equation of the tangent at a point $P$ on the parabola
$y^{2} = 4 a x$ the normal at which is at a distance
$a \sqrt{5} / 4$ from the focus of the parabola is

Moderate

A

$4 x - 2 y + a = 0$
B

$4 x - 8 y + 9 a = 0$
C

$2 x - y - 12 a = 0$
D

$2 x + y - 12 a = 0$

Solution

Equation of the normal with slop $m$ is
$y = m x - 2 a m - a m^{3}$ whose distance from the focus
$\begin{array}{l} (a, 0) is \frac{a m - 2 a m - a m^{3}}{\sqrt{1 + m^{2}}} = \frac{\sqrt{5}}{4} a \\ \Rightarrow 16 m^{2} (1 + m^{2}) = 5 \\ \Rightarrow m^{2} = 1 / 4 \Rightarrow m = \pm 1 / 2 \end{array}$
Slope of the tangent is $\pm 2$ and the equation of the
tangent is
$y = \pm 2 x \pm a / 2 \Rightarrow 4 x \pm 2 y + a = 0$ .

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