Question

The locus of the middle points of all chords of the parabola $y^2=4ax$ passing through the vertex is

• A. a straight line
• B. an ellipse
• C. a parabola
• D. a circle

Answer 1

The correct option is C a parabola

Let $OB$ be chord and let $B(h,k)$
Then, coordinate of $M$ b $(\frac{h}{2},\frac{k}{2})$.
But $(h,k)$ lies on the parabola
$\therefore$ $k^2=4ah$
$\Rightarrow$ ${\left(\frac{k}{2}\right)}^2\times 4=8a\frac{h}{2}$
Hence, locus of $(\frac{h}{2},\frac{k}{2})$ is $y^2=2ax$
Hence, it represents a parabola.