First slide

Parabola

Question

The point of intersection of the normals to the parabola $y^{2} = 4 x$ at the ends of its latus rectum is

Moderate

A

(0, 2)
B

(3, 0)
C

(0, 3)
D

(2, 0)

Solution

Equation of the latus rectum is $x = 1$ and the coordinates of the ends of the latus rectum are $(1, 2)$ and $(1, - 2) .$
Equations of the normal at these points is $y = - t x + 2 a t + a t^{3}$ , where $t = \pm 1, a = 1 \Rightarrow y + x = 3$ and
$y - x = - 3$ which intersect at $(3, 0) .$

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