Question

Let $PM$ be the perpendicular from the point $P(1,2,3)$ to $x-y$ plane. If $OP$ makes an angle $\theta$ with the positive direction of $z-$ axis and $OM$ makes an angle $\varphi$ with the positive direction of $x-$ axis, where $O$ is the origin, then

• A. $\tan \varphi =2$
• B. $9{\tan }^2\theta +\tan \varphi =7$
• C. Length of $OM=\sqrt{5}$
• D. $\tan \varphi =\sqrt{5}$

Answer 1

Answer: (A), (B), (C)

Length of $OM=\sqrt{5}$

We know that,
$r=\sqrt{1^1+2^2+3^2}=\sqrt{14}$
$OM=r\sin \theta 3=r\cos \theta \Rightarrow \cos \theta =\frac{3}{\sqrt{14}}$
$\therefore OM=\sqrt{14}\times \sqrt{\frac{5}{14}}=\sqrt{5}$
$1=r\sin \theta \cos \varphi \cdots \left(1\right)2=r\sin \theta \sin \varphi \cdots \left(2\right)\Rightarrow \tan \varphi =2$
$9{\tan }^2\theta +\tan \varphi =9\times {\left(\frac{\sqrt{5}}{3}\right)}^2+2\Rightarrow 9{\tan }^2\theta +\tan \varphi =7$

Alternate Solution:
Any point $(x,y,z)$ in $3D$ plane can be represented as
$x=r\sin \theta \cos \varphi$
$y=r\sin \theta \sin \varphi$
$z=r\cos \theta$
For point $P(1,2,3)$,
$1=r\sin \theta \cos \varphi ...\left(1\right)$
$2=r\sin \theta \sin \varphi ...\left(2\right)$
$3=r\cos \theta ...\left(3\right)$
$\left(2\right)÷\left(1\right)$
$\tan \varphi =2$
Squaring and adding eqn $\left(1\right)$ and $\left(2\right)$
$5=r^2{\sin }^2\theta$
$\Rightarrow r\sin \theta =\sqrt{5}=OM(\because \theta \text{is +ve})$

$\frac{r\sin \theta }{r\cos \theta }=\frac{\sqrt{5}}{3}$ from eqn $\left(3\right)$
$\Rightarrow \tan \theta =\frac{\sqrt{5}}{3}$
$9{\tan }^2\theta +\tan \varphi =9\times {\left(\frac{\sqrt{5}}{3}\right)}^2+2\Rightarrow 9{\tan }^2\theta +\tan \varphi =7$