Question

The combined equation of a pair of tangents to a circle drawn from the origin $O$ be $xy-y^2=(2+\sqrt{3})\left(x^2-xy\right)$. The radius of the circle is $3$ units and the centre is in first quadrant. Evaluate $(2-\sqrt{3})|OA|$ where $A$ is one of the points of contact.

Answer 1

Answer: 3

$xy-y^2=(2+\sqrt{3})\left(x^2-xy\right)$
$\iff y(x-y)=(2+\sqrt{3})(x)(x-y)$
$\iff (x-y)[(2+\sqrt{3})x-y]=0$
$\iff x=y$ or $y=2+\sqrt{3}x$
(Both equations represent lines with inclinations $45^{\circ }$ and $75^{\circ }$ )

$\angle BOA=30^{\circ }$
$\Rightarrow \angle COA=15^{\circ }$
$|OA|=|CA|\cot 15^{\circ }$
$=\frac{3}{2-\sqrt{3}}$
$\Rightarrow (2-\sqrt{3})|OA|=3$