Question

Let $p_1: x-2 y+3 z=5$ and $p_2: 2 x+3 y+z+4=0$ be two planes. If $P$ is the foot of the perpendicular dropped from the origin ' $O$ ' to the line of intersection of the planes, then

• A. vectors $\overrightarrow{O P}, \hat{i}-2 \hat{j}+3 \hat{k}$ and $2 \hat{i}+3 \hat{j}+\hat{k}$ are coplanar.
• B. If $Q$ is another point on the line of intersection of planes then equation of the plane containing a triangle $O P Q$ is $14 x +7 y +17 z =0$.
• C. Equation of the plane perpendicular to the line of intersection of the planes and passing through $(1,1,1)$ is $11 x -5 y -7 z +1=0$.
• D. If $N _1 \& N _2$ are foot of the perpendicular from the origin ' $O$ ' to the planes $p _1 \& P _2$ then $ON _1+ ON _2$ is equal to $\frac{9}{\sqrt{14}}$.

Answer 1

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

(A) Obviously true.
(B) $p _1+\lambda p _2=0$ which passes through $(0,0,0)$
(C) Vector along the line of intersection of planes is
$\begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 1 & -2 & 3 \\ 2 & 3 & 1\end{vmatrix}=-11 \hat{ i }+5 \hat{ j }+7 \hat{ k }$
$\therefore-11(x-1)+5(y-1)+7(z-1)=0 $
$11 x-5 y-7 z+1=0$
(D) $ ON _1+ ON _2=\frac{5}{\sqrt{14}}+\frac{4}{\sqrt{14}}=\frac{9}{\sqrt{14}}$