If congruent tangents of $x^2 + y^2 = r^2$ and $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ form a square, then the length of diagonal of the square is
Step-by-Step Solution
Key Concept: The director circle of an ellipse and inscribed figures relate through geometric properties; the diagonal of a square inscribed in a circle equals $\sqrt{2}$ times the side length.
The director circle of $x^2 + y^2 = r^2$ and $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ represents the same circle. For a square inscribed in this circle with side $2r - 6\sqrt{2}$, the diagonal equals $\sqrt{2} \times \text{side} = \sqrt{2}(2r - 6\sqrt{2}) = 10$. Solving: $2r\sqrt{2} - 12 = 10$, so $2r\sqrt{2} = 22$, giving $r\sqrt{2} = 11$. The diagonal of the square is $\sqrt{2} \times \text{side} = 10$.
Correct Answer: 10