Question

Let $P$ be a point in the first octant, whose image $Q$ in the plane $x+y=3$(that is, the line segment $PQ$ is perpendicular to the plane $x+y=3$ and the mid-point of $PQ$ lies in the plane $x+y=3$) lies on the z-axis. Let the distance of $P$ from the x-axis be $5$. If $R$ is the image of $P$ in the xy-plane, then the length of $PR$ is _____ .

Answer 1

Answer: 8

Let $P(\alpha ,\beta ,\gamma )Q(0,0,\gamma )\&R(\alpha ,\beta ,-\gamma )$
Now $\overrightarrow{PQ}||\hat{i}+\hat{j}\Rightarrow (\alpha \hat{i}+\beta \hat{j})||(\hat{i}+\hat{j})\Rightarrow \alpha =\beta$
Also, mid point of $PQ$ lies on the plane
$\Rightarrow \frac{\alpha }{2}+\frac{\beta }{2}=3$
$\Rightarrow \alpha +\beta =6\Rightarrow \alpha =3$
Now, distance of point $P$ from $X$-axis
$\sqrt{{\beta }^2+{\gamma }^2}=5$
$\Rightarrow {\beta }^2+{\gamma }^2=25$
$\Rightarrow {\gamma }^2=16$ as $\beta =\alpha =3$ as $\gamma =4$
Hence $PR=2\gamma =8$