Question

If $e_1$ and $e_2$ are the eccentricities of the hyperbola $xy=9$ and $x^2-y^2=25,$ then $(e_1,e_2)$ lie on a circle $C_1$ with centre origin then the $($radius${)}^2$ of the director circle of $C_1$ is

• A. $2$
• B. $4$
• C. $8$
• D. $16$

Answer 1

The correct option is C $8$

$e_1=\sqrt{2},e_2=\sqrt{2}$
$\Rightarrow (\sqrt{2},\sqrt{2})$ is the point on the circle.
$\Rightarrow$ Radius of $C_1=\sqrt{{\sqrt{2}}^2+{\sqrt{2}}^2}=2.$
$\Rightarrow$ Radius of director circle of $C_1=2\sqrt{2}$
$\therefore ($Radius${)}^2=8$
Hence, option 'B' is correct.