First slide

Parabola

Question

If $P$ be the point $(1, 0)$ and $Q$ , a point on the locus $y^{2} = 8 x$ . The locus of the mid point of $P Q$ is:

Moderate

A

$x^{2} + 4 y + 2 = 0$
B

$x^{2} - 4 y + 2 = 0$
C

$y^{2} - 4 x + 2 = 0$
D

$y^{2} + 4 x + 2 = 0$

Solution

Let the coordinates of $Q$ be $(2 t^{2}, 4 t)$ coordinates of
the mid point of PQ are $(\frac{2 t^{2} + 1}{2}, 2 t)$
Let $h = \frac{2 t^{2} + 1}{2}, k = 2 t$ Eliminating t, we get $4 h = k^{2} + 2$
So the required locus is $y^{2} - 4 x + 2 = 0$

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