Question

PQ is a double ordinate of the ellipse x² + 9y² = 9, the normal at P meets the diameter through Q at R, then the locus of midpoint of PR is–

• A. A circle
• B. A parabola
• C. An ellipse
• D. A hyperbola

Answer 1

Answer: (C)

An ellipse

Answer 2

$\frac{x^{2}}{9}$+ $\frac{y^{2}}{1}$= 1

P ŗ (3 cos q, sin q), Q = (3 cos q, – sin q)

Equation of diameter CQ

y = – $\left( \frac{1}{3}\tan\theta \right)$x ...(1)

Normal at P

$\frac{3x}{\cos\theta}$– $\frac{y}{\sin\theta}$= 8 ....(2)

From (1) and (2), we get

R ŗ $\left( \frac{12}{5}\cos\theta, - \frac{4}{5}\sin\theta \right)$

Let mid-point of PR be

M ŗ $\left( \frac{27}{10}\cos\theta,\frac{1}{10}\sin\theta \right)$ŗ (h, k)

cos q = $\frac{10h}{27}$, sin q = 10 k So locus $\left( \frac{10x}{27} \right)^{2}$+ (10y)² = 1

Ž which is an ellipse