Question

Let $x^{2}+y^{2}=4r^{2}$ and $xy=1$ intersects at $A$ and $B$ in first quadrant. If $AB=\sqrt{14}$ units, then the value of $\left|2 r\right|$ is

Answer 1

Let $A=t, \frac{1}{t} \& B=\frac{1}{t}, t$
$\because$ Both curves are symmetric about line $y=x$
then, $A B^2=t-\frac{1}{t}^2+\frac{1}{t}-t^2=14$
$\Rightarrow 2 t^2+\frac{1}{t^2}=14+4 $
$\Rightarrow t^2+\frac{1}{t^2}=9$
Now, $O A^2=t^2+\frac{1}{t^2}=9 \Rightarrow 2 r^2=9$
$\Rightarrow 2 r=3$