Line $L_1$ passes through the point $(1,2,3)$ and is parallel to Z -axis. Line $\mathrm{L}_2$ passes through the point $(\lambda, 5,6)$ and is parallel to $y$-axis. Let for $\lambda=\lambda_1, \lambda_2, \lambda_2 \lt \lambda_1$, the shortest distance between the two lines be 3 . Then the square…

Question

Line $L_1$ passes through the point $(1,2,3)$ and is parallel to Z -axis. Line $\mathrm{L}_2$ passes through the point $(\lambda, 5,6)$ and is parallel to $y$-axis. Let for $\lambda=\lambda_1, \lambda_2, \lambda_2 \lt \lambda_1$, the shortest distance between the two lines be 3 . Then the square of the distance of the point $\left(\lambda_1, \lambda_2, 7\right)$ from the line $\mathrm{L}_1$ is

Aicharya · Accepted Answer

$\begin{aligned} & L_1 \equiv \frac{x-1}{0}=\frac{y-2}{0}=\frac{z-3}{1} \\ & L_2 \equiv \frac{x-\lambda}{0}=\frac{y-5}{1}=\frac{z-6}{0}\end{aligned}$
$\mathrm{SD}=\frac{\left|\begin{array}{ccc}\lambda-1 & 3 & 3 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right|}{\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right|}$
$=|\lambda-1|=3$
$\begin{aligned}
& \lambda=4,-2 \\ & \lambda_1=4 \\ & \lambda_2=-2
\end{aligned}$
Let foot of perpendicular from
$\begin{aligned}
& \mathrm{P}(4,-2,7) \text { is } \mathrm{Q}(1,2, \mathrm{t}+3) \\ & \mathrm{So}(3,-4,4-\mathrm{t}) \cdot(0,0,1)=0 \\ & \mathrm{t}=4
\end{aligned}$
$\begin{aligned}
& \mathrm{So} \mathrm{Q}(1,2,7) \\ & \mathrm{PQ}^2=9+16 \\ & \mathrm{PQ}^2=25
\end{aligned}$ ^

Line $L_1$ passes through the point $(1,2,3)$ and is parallel to Z -axis. Line $\mathrm{L}_2$ passes through…

Solution