Question

The locus of the mid point of the line segment joining the focus to a moving point on the parabola $y^2$=4ax is another parabola with directrix

• A. $x=-a$
• B. $x=-\frac{a}{2}$
• C. $x=0$
• D. $x=\frac{a}{2}$

Answer 1

Answer: (C)

$x=0$

Given a moving point on parabola ,$y^2=4ax$

Consider any point on the parabola $(a{t}^2,2at)$

$\therefore$ The mid point $(h,k)=(\frac{a{t}^2+a}{2},\frac{2at+0}{2})$

$\therefore 2h=a(t^2+1),at=K2h=a(\frac{K^2}{a^2}+1)\therefore Locus\Rightarrow 2xa=y^2+a^2\therefore y^2-2a(x-\frac{a}{2})$

$\therefore$ Direction :$x=\frac{-a}{2}+\frac{a}{2}=0$ or y-axis