Question

Equation of the plane through the mid-point of the line segment joining the points $P(4,5,-10),Q(-1,2,1)$ and perpendicular to $PQ$ is

• A. $r.(\frac{}{3}2\hat{i}+\frac{7}{2}\hat{j}-\frac{9}{2}\hat{k})=45$
• B. $r.(-\hat{i}+2\hat{j}+\hat{k})=\frac{135}{2}$
• C. $r.(5\hat{i}+3\hat{j}-11\hat{k})+\frac{135}{2}=0$
• D. $r.(4\hat{i}+5\hat{j}-10\hat{k})=85$
• E. $r.(5\hat{i}+3\hat{j}-11\hat{k})=\frac{135}{2}$

Answer 1

The correct option is E $r.(5\hat{i}+3\hat{j}-11\hat{k})=\frac{135}{2}$

Coordinate the mid-point of $P(4,5,-10)$ and $Q(-1,2,1)$ is
$(\frac{4-1}{2},\frac{5+2}{2},\frac{-10+1}{2})$. i.e. $(\frac{3}{2}.\frac{7}{2},\frac{-9}{2})$
Now, DR's of $PQ$ is $(-1-4,2-5,1+10)$
i.e., $(-5,-3,11)$ or $(5,3,-11)$
Therefore, euation of plane passing through
$(\frac{3}{2},\frac{7}{2},\frac{-9}{2})$ and having DR's $(5,3,-11)$ is
$5(x-\frac{3}{2})+3(y-\frac{7}{2})-11(z-\frac{9}{2})=0$
$\Rightarrow 5x+3y-11z=\frac{15}{2}+\frac{21}{2}+\frac{99}{2}$
$\Rightarrow 5x+3y-11z=\frac{135}{2}$
It is written by vector form
$r.(5\hat{i}+3\hat{j}-11\hat{k})=\frac{135}{2}$