Question

The combined equation of a pair of tangents to a circle drawn from the origin $O$ be $x y-y^{2}=(2+\sqrt{3})\left(x^{2}-x y\right)$. 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

$x y-y^{2}=(2+\sqrt{3})\left(x^{2}-x y\right)$
$\Leftrightarrow y(x-y)=(2+\sqrt{3})(x)(x-y)$
$\Leftrightarrow(x-y)[(2+\sqrt{3}) x-y]=0$
$\Leftrightarrow x=y$ 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}$
$| OA |=| CA | \cot 15^{\circ}$
$=\frac{3}{2-\sqrt{3}}$
$\Rightarrow(2-\sqrt{3})| OA |=3$