First slide

Parabola

Question

Tangents at the extremities of a focal chord of a
parabola intersect

Moderate

A

on the axis of the parabola
B

on the tangent at the vertex
C

at the point of intersection of the directrix and the
line parallel to the axis of the parabola through
the mid-point of the chord
D

none of these

Solution

Extremities of a focal chord are $(a t^{2}, 2 a t), (a / t^{2}, - 2 a / t)$
Tangents are $t y = x + a t^{2}, and - t y = x t^{2} + a$
They intersect at $(- a, \frac{a (t^{2} - 1)}{t}) or (- a, a (t - \frac{1}{t}))$
which is the point of intersection of the directrix
$x = - a$ and the line $y = a (t - \frac{1}{t})$ which is the
line through the mid-point of the chord parallel to
the axis.

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