Question

The locus of the middle point of the intercept of the tangents drawn from an external point to the ellipse x² + 2y² = 2 between the coordinates axes, is-

• A. $\frac{1}{x^{2}}$+ $\frac{1}{2y^{2}}$= 1
• B. $\frac{1}{4x^{2}}$+ $\frac{1}{2y^{2}}$= 1
• C. $\frac{1}{2x^{2}}$+ $\frac{1}{4y^{2}}$= 1
• D. $\frac{1}{2x^{2}}$+ $\frac{1}{y^{2}}$= 1

Answer 1

Answer: (C)

$\frac{1}{2x^{2}}$+ $\frac{1}{4y^{2}}$= 1

Answer 2

$\frac{x^{2}}{2} + \frac{y^{2}}{1} = 1$ … (1)

\ Eqn. of tangent AB at point P

̃$\frac{x}{\sqrt{2}}$cos q +$\frac{y}{1}$sinq =1

̃ A($\sqrt{2}$sec q, 0) & B (0, cosec q) Let M is the middle point,

\ M (h = $\frac{\sqrt{2}\sec\theta + 0}{2}$, k = $\frac{0 + \text{cosec}\theta}{2}$)

̃ cos q = $\frac{1}{h\sqrt{2}}$ & sin q = $\frac{1}{2k}$

̃ cos² q + sin²q = 1 ̃ $\frac{1}{2x^{2}}$+$\frac{1}{4y^{2}}$= 1