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  • If the shortest distance between the lines <img alt="\frac{x+\sqrt{6}}{2}=\frac{y-\sqrt{6}}{3}=\frac{z-\sqrt{6}}{4}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cfrac%7Bx&plus;%

If the shortest distance between the lines \frac{x+\sqrt{6}}{2}=\frac{y-\sqrt{6}}{3}=\frac{z-\sqrt{6}}{4}  and \frac{x-\lambda}{3}=\frac{y-2 \sqrt{6}}{4}=\frac{z+2 \sqrt{6}}{5} is 6, 

then the square of sum of all possible values of \lambda is

Option: 1

389


Option: 2

"__"


Option: 3

"__"


Option: 4

"__"


Answers (1)

best_answer

\begin{aligned} & \mathrm{P}(-\sqrt{6}, \sqrt{6}, \sqrt{6}) \quad \mathrm{Q}(\lambda, 2 \sqrt{6},-2 \sqrt{6}) \\ & \overline{\mathrm{n}}_1=(2,3,4) \quad \overline{\mathrm{n}}_2=(3,4,5) \\ & \overline{\mathrm{n}}_1 \times \overline{\mathrm{n}}_2 \Rightarrow\left|\begin{array}{ccc} \hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{array}\right|=\hat{\mathrm{i}}(-1)-\hat{\mathrm{j}}(-2)+\hat{k}(-1) \\ & \end{aligned}

\begin{aligned} & =(-1,2,-1) \\ & \therefore \mathrm{S}_{\mathrm{d}}\left|\frac{\overline{\mathrm{PQ}} \cdot(-1,2,-1)}{\sqrt{6}}\right|=\frac{(\lambda+\sqrt{6}, \sqrt{6},-3 \sqrt{6}) \cdot(-1,2,-1)}{\sqrt{6}} \\ & =\left|\frac{-\lambda-\sqrt{6}+2 \sqrt{6}+3 \sqrt{6}}{\sqrt{6}}\right|=6 \end{aligned}

\begin{aligned} & \Rightarrow|-\lambda+4 \sqrt{6}|=6 \sqrt{6} \\ & \text { (+) }-\lambda+4 \sqrt{6}=6 \sqrt{6} \\ & \text { (-) } \lambda-4 \sqrt{6}=6 \sqrt{6} \\ & \lambda=-2 \sqrt{6} \\ & \lambda=10 \sqrt{6} \\ & \therefore(8 \sqrt{6})^2=384 \\ & \end{aligned}

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Suraj Bhandari

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