`P(x , y)` is a variable point on the parabola `y^2=4a x` and `Q(x+c ,y+c)` is another variable point, where `c` is a constant. The locus of the midpoint of `P Q` is an (a)parabola (b) ellipse (c)hyperbola (d) circle

Options

A. parabola
B. ellipse
C. hyperbola
D. circle

Answer

A

Explanation

(1) Let the midpoint of PQ be `(alpha,beta)`. Then,
`alpha=x+(c)/(2)andbeta=y+(c)/(2)`
`or(beta-(c)/(2))^(2)=4a(alpha-(c)/(2))`
`or(y-(c)/(2))^(2)=4a(x-(c)/(2))`
which is the required locus.