Question

The reflection of $A (\sqrt{3}, 1)$ on the line $y=x$ is point $P$. The segment $OP$ is rotated by $15^{\circ}$ in clockwise direction ( $O$ is the origin) so that point $P$ reaches to point $Q (k, m)$. Evaluate $k^{2}+m^{2}$.

Answer 1

$A=(\sqrt{3}, 1)$ and $P$ is the reflection of point $A$ on the line $y=x$.
$\Rightarrow P =(1, \sqrt{3})$
$\Rightarrow OM =1, MP =\sqrt{3}$
$\Rightarrow OP =\sqrt{1+3}=2$
$\Rightarrow \angle POM =60^{\circ}$

$\angle POQ =15^{\circ}, \angle POM =60^{\circ} $
$\Rightarrow \angle QON =45^{\circ}$
$\Rightarrow Q$ lies on the line $y=x$
Let $Q =(h, h)$
$OP = OQ =2 $
$\Rightarrow \sqrt{h^{2}+h^{2}}=2$
$\Rightarrow h=\sqrt{2}$
$\Rightarrow k=m=\sqrt{2}$
$ \Rightarrow k^{2}+m^{2}=4$