Question
Let $$P$$ be the point $$\left( {1,\,0} \right)$$ and $$Q$$ a point on the locus $${y^2} = 8x.$$ The locus of mid point of $$PQ$$ is :
A.
$${y^2} - 4x + 2 = 0$$
B.
$${y^2} + 4x + 2 = 0$$
C.
$${x^2} + 4y + 2 = 0$$
D.
$${x^2} - 4y + 2 = 0$$
Answer :
$${y^2} - 4x + 2 = 0$$
Solution :
$$P = \left( {1,\,0} \right)\,\,Q = \left( {h,\,k} \right)$$ Such that $${K^2} = 8h$$
Let $$\left( {\alpha ,\,\beta } \right)$$ be the midpoint of $$PQ$$
$$\eqalign{
& \alpha = \frac{{h + 1}}{2},\,\,\,\,\,\,\,\,\,\,\beta = \frac{{k + 0}}{2}\, \cr
& 2\alpha - 1 = h\,\,\,\,\,\,\,\,\,\,2\beta = k \cr
& {\left( {2\beta } \right)^2} = 8\left( {2\alpha - 1} \right)\,\, \Rightarrow {\beta ^2} = 4\alpha - 2 \cr
& \Rightarrow {y^2} - 4x + 2 = 0 \cr} $$