From a point $P$, perpendicular tangents are drawn to the ellipse $x^2 + 2y^2 = 2$. If the chords of contact are tangents to a family of concentric circles, having the centres same as that of the ellipse then the ratio of the areas of the largest circle to the smallest circle is____.
Step-by-Step Solution
Key Concept: The locus of point P from which perpendicular tangents are drawn to ellipse x² + 2y² = 2 is the director circle x² + y² = 3. The chord of contact from P(√3cosθ, √3sinθ) is tangent to concentric circles, with varying radius determined by a = 2/√(3cos²θ + 12sin²θ), requiring optimization to find the ratio of maximum to minimum circle areas.
Point $P$ lies on director circle $x^2 + y^2 = 3$. Let $P = (\sqrt{3}\cos\theta, \sqrt{3}\sin\theta)$. The chord of contact has length $a\sqrt{3}\cos\theta + 2\sqrt{3}a\sin\theta = 2$. For tangency to circle $x^2 + y^2 = a^2$, this gives $a = \frac{2}{\sqrt{3\cos^2\theta + 12\sin^2\theta}}$. Maximizing yields ratio $\frac{\max \text{ area}}{\min \text{ area}} = 4$.
Correct Answer: 4