Question

Let the plane $P:\vec{r}\cdot \vec{a}=d$ contain the line of intersection of two planes $\vec{r}\cdot (\hat{i}+3\hat{j}-\hat{k})=6$ and $\vec{r}\cdot (-6\hat{i}+5\hat{j}-\hat{k})=7$. If the plane $P$ passes through the point $\left(2,3,\frac{1}{2}\right)$, then the value of $\frac{|13\vec{a}{|}^2}{d^2}$ is equal to

• A. 90
• B. 93
• C. 95
• D. 97

Answer 1

Answer: (B)

Equation of plane passing through line of intersection of planes
$P_1:\vec{r}((\hat{i}+3\hat{j}-\hat{k})=6$ and $P_2:r\cdot (-6\hat{i}+5\hat{j}-\hat{k})=7$ is
$P_1+\lambda P_2=0$
$(\overline{r}\cdot (\hat{i}+3\hat{j}-\hat{k})-6)+\lambda (\overline{r}\cdot (-6\hat{i}+5\hat{j}-\hat{k})-7)=0$
and it passes through point $\left(2,3,\frac{1}{2}\right)$
$\Rightarrow \left(2+9-\frac{1}{2}-6\right)+\lambda \left(-12+15-\frac{1}{2}-7\right)=0$
$\Rightarrow \lambda =1$
Equation of plane is $\overline{r}\cdot (-5\hat{i}+8\hat{j}-2\hat{k})=13$
$|\vec{a}{|}^2=25+64+4=93;d=13$
Value of $\frac{|13\bar{a}{|}^2}{d^2}=93$