First slide

Parabola

Question

If $P$ is the point $(1, 0)$ and $Q$ lies on the parabola
$y^{2} = 36 x$ then the locus of the mid point of $P Q$ is

Moderate

A

$y^{2} = 9 (2 x - 1)$
B

$y^{2} = 9 (x + 2)$
C

$y^{2} = 2 (x - 9)$
D

$x^{2} = 9 (y - 2)$

Solution

Let the coordinates of $Q$ be $(9 t^{2}, 18 t)$ coordinates
of $R$ the mid point of $P Q$ are $(\frac{9 t^{2} + 1}{2}, 9 t) = (x, y)$
Locus of $R$ is
$\begin{array}{l} 2 x = 9 {(\frac{y}{9})}^{2} + 1 \\ \Rightarrow y^{2} = 9 (2 x - 1) \end{array}$

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