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Q.
If $P$ is $(3,1)$ and $Q$ is a point on the curve $y^{2}=8 x$, then the locus of the mid-point of the line segment $PQ$ is
TS EAMCET 2018
Solution:
$P(3,1)$ and $Q$ is a point of parabola $y^{2}=8 x$
Let $R_{(h, k)}$ is the mid-point of $P Q$.
So, coordinate of $Q(2 h-3,2 k-1)$ point $Q$ lie on parabola, so $(2 k-1)^{2}=8(2 h-3)$
$\Rightarrow \,4 k^{2}-4 k+1=16 h-24$
$\Rightarrow \, 4 k^{2}-4 k-16 h+25=0$
Hence, locus of $R(h, k)$ is $4 y^{2}-4 y-16 x+25=0$
