First slide

Parabola

Question

If the normal drawn from the point on the axis
of the parabola $y^{2} = 8 a x$ whose distance from the
focus is 8 $a,$ and which is not parallel to either
axes, makes an angle $θ$ with the axis of $x,$ then $θ$
is equal to

Moderate

A

$π / 6$
B

$π / 4$
C

$π / 3$
D

none of these

Solution

Focus is $(2 a, 0),$ so point is $P (2 a + 8 a, 0)$ Equation
of any normal is $y = m x - 2 (2 a) m - (2 a) m^{3}$ . If it
passes through $P (10 a, 0)$ , then
$6 a m - 2 a m^{3} = 0 \Rightarrow m = 0, \pm \sqrt{3}$
So the required angle is $θ = \tan^{- 1} \sqrt{3} = π / 3$ .

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