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. A hyperbola

Answer 1

Answer: (C)

A parabola

Answer 2

Let y² = 4ax ….(i)

Q PM = QM

Q $\left\{ \begin{matrix} \angle PAQ = \frac{\pi}{2} \\ \therefore t_{1}t_{2} = - 4 \end{matrix} \right.\$

M $\left\{ \begin{matrix} h = \frac{at_{1}^{2} + at_{2}^{2}}{2}........(ii) \\ k = \frac{2at_{1} + 2at_{2}}{2}.......(iii) \end{matrix} \right.\$

By (iii); k² = a² (t₁ + t₂)²

= a² (t₁² + t₂² + 2t₁t₂)

= $a^{2}\left\{ \frac{2h}{a} + 2( - 4) \right\}$

̃ $y^{2} = a^{2}\left( \frac{2x}{a} \right) - 8a^{2}$

̃ y² = 2ax – 8a²

It’s a equation of parabola