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

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 $P Q$ 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 $P R=2 \gamma=8$