Question

Let $P$ be a point in the first octant, whose image $Q$ 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 $P\left(\alpha , \, \beta , \, \gamma \right)$
$Q\left(0,0 , \gamma \right)$ and
$R\left(\alpha , \, \beta , \, - \gamma \right)$
Now, $\overrightarrow{P Q}\parallel\hat{i}+\hat{j}\Rightarrow \alpha \hat{i}+\beta \hat{j}\parallel\hat{i}+\hat{j}$
$\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 is $\sqrt{\beta ^{2} + \gamma ^{2}}=5$
$\Rightarrow , \beta ^{2}+\gamma ^{2}=25\Rightarrow \gamma ^{2}=16$
As $\beta =\alpha =3$
As $\gamma =4$
Hence, $PR=2\gamma =8$