Question

If common tangents of $x^2+y^2=r^2$ and $\frac{x^2}{16}+\frac{y^2}{9}=1$ form a square, then the length of the diagonal of the square is

• A. $10$
• B. $25$
• C. $5r$
• D. $5\sqrt{2}r$

Answer 1

The correct option is A $10$


Common tangents of the curves form a square.
$\Rightarrow$ Tangents are perpendicular.
So, director circle of $C:x^2+y^2=r^2$ and $E:\frac{x^2}{16}+\frac{y^2}{9}=1$ is same.
Director circle of circle $C$ is $x^2+y^2=2r^2.$
Director circle of ellipse $E$ is $x^2+y^2=16+9=25$
$\therefore 2r^2=25$
$\Rightarrow r=\frac{5}{\sqrt{2}}$
$\therefore$ Side of square $=2r=5\sqrt{2}$
$\Rightarrow$ Diagonal of square $=\sqrt{2}\text{(Side of square)}=10$