First slide

Parabola

Question

$P$ is a point on the locus of the mid-points of the
chords of the parabola $y^{2} = 4 a x$ passing through
the vertex of the parabola; $S$ is the focus of the
parabola and the line joining $S$ and $P$ is produced
to meet the parabola at $Q$ . If the ordinate of $P$ is
equal to its abscissa, coordinates of $Q$ are

Moderate

A

$(4 a, 4 a)$
B

$(3 a, 3 a)$
C

$(2 a, 2 a)$
D

none of these

Solution

$O (0, 0)$ is the vertex and $R (a t^{2}, 2 a t)$ is any point
on the parabola. $(a t^{2} / 2, 2 a t / 2)$ is the mid-point of
the chord $O R$ . Whose locus is $y^{2} = 2 a x$ coordinates
of P on this locus are $(2 a, 2 a)$ , focus is
$S (a, 0) .$ Equation of $S P$ is $y = 2 (x - a)$ which meets
$y^{2} = 4 a x$ at the point $Q$ for which $4 (x - a)^{2} = 4 a x$
$\Rightarrow x^{2} - 3 a x + a^{2} = 0$ which is not satisfied by the
points in $(a), (b) o r (c) .$

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