If the shortest between the lines x + 6 2 = y - 6 3 = z - 6 4 and x - 3 = y - 2 6 4 = z + 2 6 5 is 6, then the square of sum of all possible values of is :

Shortest distance between the lines x+62=y-63=z-64 x-3=y-2 64=2+2 65 is 6 Vector along line of shortest distance =|arrayllli & j & k \\ 2 & 3 & 4 \\ 3 & 4 & 5array|, -i+2 j-k (its magnitude is 6 ) Now 16|arrayccc6+ & 6 & -3 6 \\ 2 & 3 & 4 \\ 3 & 4 & 5array|= 6 =-2 6, 10 6 So, sq

If the shortest between the lines x 6 2 y 6 3 z 6 4 and x la: null (Mathematics)

Shortest distance between the lines

\frac{x+\sqrt{6}}{2}=\frac{y-\sqrt{6}}{3}=\frac{z-\sqrt{6}}{4}

\frac{x-\lambda}{3}=\frac{y-2 \sqrt{6}}{4}=\frac{2+2 \sqrt{6}}{5}

is 6

Vector along line of shortest distance

=\left|\begin{array}{lll}\mathrm{i} & \mathrm{j} & \mathrm{k} \\ 2 & 3 & 4 \\ 3 & 4 & 5\end{array}\right|, \Rightarrow-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\mathrm{k}

(its magnitude is

\sqrt{6}

)

Now

\frac{1}{\sqrt{6}}\left|\begin{array}{ccc}\sqrt{6}+\lambda & \sqrt{6} & -3 \sqrt{6} \\ 2 & 3 & 4 \\ 3 & 4 & 5\end{array}\right|=\pm 6

\Rightarrow \lambda=-2 \sqrt{6}, 10 \sqrt{6}

So, square of sum of these values

=(10 \sqrt{6}-2 \sqrt{6})^2=(8 \sqrt{6})^2=384

Explanation

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