Question

Consider two fixed points $A (1,0,0)$ and $P (1,0,1)$. If $OA$ (where ' $O$ ' is origin) is rotated through ' $O$ ' in xy-plane by a variable angle $\theta$, to reach a new position $OB$, then which of the following statement(s) is(are) correct?

• A. Maximum volume of tetrahedron OAPB is $\frac{1}{6}$.
• B. Maximum volume of tetrahedron $OAPB$ is unity.
• C. The distance of plane OPB from point A is $\frac{|\sin \theta|}{\sqrt{\left(1+\sin ^2 \theta\right)}}$.
• D. Equation of the plane OPB is $(z-x) \sin \theta+y \cos \theta=0$.

Answer 1

Answer: (A), (C), (D)

Volume of tetrahedron $OAPB =\frac{1}{6}|[\overrightarrow{ OA } \overrightarrow{ OB } \overrightarrow{ OP }]|$
$=\frac{1}{6}\begin{vmatrix}1 & 0 & 0 \\ \cos \theta & \sin \theta & 0 \\ 1 & 0 & 1\end{vmatrix}|=| \frac{\sin \theta}{6} \mid$

Also, $h=\overrightarrow{ OP } \times \overrightarrow{ OB }=\begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 1 & 0 & 1 \\ \cos \theta & \sin \theta & 0\end{vmatrix}=-\sin \theta \hat{ i }-\hat{ j }(-\cos \theta)+\hat{ k }(\sin \theta)$
As, $r.h =0 \Rightarrow- x \sin \theta+ y \cos \theta+ z \sin \theta=0$
$d=\frac{|-\sin \theta|}{\sqrt{\left(\sin ^2 \theta+\cos ^2 \theta+\sin ^2 \theta\right)}}=\frac{|\sin \theta|}{\sqrt{\left(1+\sin ^2 \theta\right)}}$