Question

The locus of the middle points of chords of a parabola which subtend a right angle at the vertex of the parabola is

• A. a circle
• B. an ellipse
• C. a parabola
• D. none of these

Answer 1

The correct option is C a parabola

let $(h,k)$ be the middle point of chord of parabola $y^2=4ax$ ...(1)
Then equation of chord of contact is $T=S_1$
$\Rightarrow yk-2a(x+h)=k^2-4ah$
$\Rightarrow 2ax-yk=2ah-k^2$ ...(2)
homogenising (2) wrt (1): $y^2=4ax\left(\frac{2ax-yk}{2ah-k^2}\right)$
Since, chord subtends right angle at origin
Therefore, co-efficient of $x^2=$coefficient of $y^2$
$2ah-k^2=-8a^2$
Therefore, locus is $y^2=2a(x+4a)$ which is a parabola.