Question

The locus of the middle points of chords of the parabola $y^2=8x$ drawn through the vertex is a parabola whose

• A. Focus is $(2,0)$
• B. Latusrectum $=8$
• C. Focus is $(0,2)$
• D. Latusrectum $=4$

Answer 1

Answer: (D)

Latusrectum $=4$

If the middle point of a chord is $(\alpha ,\beta )$, then
$\alpha =\frac{2t^2+0}{2},\beta =\frac{4t+0}{2}$
On eliminating $t$, we get
$\alpha ={\left(\frac{\beta }{2}\right)}^2$
Therefore, locus is $x=\frac{y^2}{4}$
$\Rightarrow y^2=4x$
This is the equation of parabola with latusrectum $4$.