Question

If a variable tangent of the circle x² + y² = 1 intersects the ellipse x² + 2y² = 4 at point P and Q, then the locus of the point of intersection of tangents at P and Q is:

• A. A circle of radius 2 units
• B. A parabola with focus as (2, 3)
• C. An ellipse with eccentricity $\frac{\sqrt{3}}{2}$
• D. None of the above

Answer 1

Answer: (C)

An ellipse with eccentricity $\frac{\sqrt{3}}{2}$

Answer 2

Let point of intersection be (h, k). Then equation of the line passing through P and Q is hx + 2ky = 4 (chord of contact). Since hx + 2ky = 4 touches x² + y² = 1, $\frac{16}{4k^{2}} = 1 + \frac{h^{2}}{4k^{2}}$ i.e., 4k² + h² = 16. So required locus is 4y² + x² = 16, which is an ellipse of eccentricity $\frac{\sqrt{3}}{2}$ and length of latus rectum is 2 units.