Question

$P$ is a point and $P M, P N$ are perpendiculars from $P$ to the $ZX$ and $X Y$ planes respectively. If $O P$ makes angles $\theta, \alpha, \beta$ $\gamma$ with the plane $O M N$ and the $X Y, Y Z, Z X$ plane respectively then $\sin ^{2} \theta\left(cosec^{2} \alpha+ cosec^{2} \beta + cosec^{2} \gamma\right)$ is equal to

• A. 0
• B. 1
• C. -1
• D. None of these

Answer 1

Answer: (B)

Let $P$ be $\left(x_{1}, y_{1}, z_{1}\right) .$ Point $M$ is $\left(x_{1}, 0, z_{1}\right)$ and $\dot{N}$ is $\left(x_{1}, y_{1}, 0\right)$
So normal to plane $O M N$ is $\overrightarrow{O M} \times \overrightarrow{O N}=\vec{x}($ say $)$
i.e. $\begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\ x_{1} & 0 & z_{1} \\ x_{1} & y_{1} & 0\end{vmatrix}$
$=\hat{i}\left(-y_{1} z_{1}\right)-\hat{j}\left(-x_{1} z_{1}\right)+\hat{k}\left(x_{1} y_{1}\right)$
therefore, $\sin \theta=\frac{-x_{1} y_{1} z_{1}+x_{1} y_{1} z_{1}+x_{1} y_{1} z}{\sqrt{x_{1}^{2}+y_{1}^{2}+z_{1}^{2}} \sqrt{\sum x_{1}^{2} y_{1}^{2}}}$
or $\left(\because \sin \theta=\frac{\vec{n} \times \overrightarrow{O P}}{|n \| \overline{O P}|}\right)$
$\Rightarrow cosec^{2} \theta=\frac{\sum x_{1}^{2} \sum x_{1}^{2} y_{1}^{2}}{\left(x_{1} y_{1} z_{1}\right)^{2}}$
$=\frac{\sum x_{1}^{2}}{x_{1}^{2}}+\frac{\sum x_{1}^{2}}{y_{1}^{2}}+\frac{\sum x_{1}^{2}}{z_{1}^{2}}$
Now, $\sin \alpha=\frac{\overrightarrow{O P} \cdot \hat{k}}{|\overrightarrow{O P}|}=\frac{z_{1}}{\sqrt{\sum x_{1}^{2}}}$
$\sin \beta=\frac{x_{1}}{\sqrt{\sum x_{1}^{2}}}$ and $\sin \gamma=\frac{y_{1}}{\sqrt{\sum x_{1}^{2}}}$
Now, $cosec^{2} \alpha + cosec^{2} \beta+ cosec^{2} \gamma$
$=\frac{x_{1}^{2}+y_{1}^{2}+z_{1}^{2}}{x_{1}^{2}}+\frac{\sum x_{1}^{2}}{y_{1}^{2}}+\frac{\sum x_{1}^{2}}{z_{1}^{2}}=cosec^{2} \theta$