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

Answer: 4

$A=(\sqrt{3},1)$ 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 $Q=(h,h)$
$OP=OQ=2$
$\Rightarrow \sqrt{h^2+h^2}=2$
$\Rightarrow h=\sqrt{2}$
$\Rightarrow k=m=\sqrt{2}$
$\Rightarrow k^2+m^2=4$