First slide

Parabola

Question

The tangent at a point $P$ on the parabola $y^{2} = 8 x$
meets the directrix of the parabola at $Q$ such that
distance of $Q$ from the axis of the parabola is 3.
Then the coordinates of $P$ cannot be

Moderate

A

(2, 4)
B

(8, 8)
C

(1/2, 2)
D

(8 – 8)

Solution

Equation of the tangent at $P (2 t^{2}, 4 t)$ to the parabola is $t y = x + 2 t^{2}$ which meets the directrix $x = - 2$ of the parabola at Q for which $y = \frac{2 t^{2} - 2}{t} .$ So that $\frac{2 t^{2} - 2}{t} = \pm 3$
. $\Rightarrow 2 t^{2} - 3 t - 2 = 0$ or $2 t^{2} + 3 t - 2 = 0$
$\Rightarrow (2 t + 1) (t - 2) = 0$ or $(2 t - 1) (t + 2) = 0$
$\Rightarrow t = \pm 2, \pm \frac{1}{2}$
So, the coordinates of P are $(8, \pm 8)$ or $(\frac{1}{2}, \pm 2)$

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