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 $\frac{|13 \bar{a}|^{2}}{ d ^{2}}=93$