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

Option: 1

90


Option: 2

93


Option: 3

95


Option: 4

97


Answers (1)

best_answer

Let the equation of Plane  \mathrm{P} is

\mathrm{(x+3 y-3-6)+\lambda(-6 x+5 y-3-7)=0}

Passes through \mathrm{\left ( 2,3,1/2 \right )}

\mathrm{\Rightarrow\left(2+9-\frac{1}{2}-6\right)+\lambda\left(-12+15-\frac{1}{2}-7\right)=0}
\mathrm{\Rightarrow \quad \lambda=1 \Rightarrow \quad P:-5 x+8 y-23-13=0}
\Rightarrow \vec{r} \cdot \left ( -5\hat{i}+8\hat{j}-2\hat{k} \right )= 13.
\Rightarrow \frac{\left | 13 \vec{a} \right |}{d^{2}}= \frac{13^{2}\left ( 5^{2}+8^{2}+2^{2} \right )}{13^{2}}= 93

Option (B)

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