First slide

Parabola

Question

If the tangents at the extremities of a focal chord of
the parabola $x^{2} = 4 a y$ meet the tangent at the vertex
at points whose abcissac are $x_{1}$ and $x_{2}$ then $x_{1} x_{2} =$

Moderate

A

$a^{2}$
B

$a^{2} - 1$
C

$a^{2} + 1$
D

$- a^{2}$

Solution

One extremity of the focal chord be $(2 a t, a t^{2})$ . Equation
of the tangent is $t x = y + a t^{2}$ which meets the
tangent at the vertex $y = 0$ at $x = a t$ so $x_{1} = a t$ and
$x_{2} = a (- \frac{1}{t})$ thus $x_{1} x_{2} = - a^{2}$ .

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