Let $P$ be the point $(10,-2,-1)$ and $Q$ be the foot of the perpendicular drawn from the point $R(1,7,6)$ on the line passing through the points $(2,-5,11)$ and $(-6,7,-5)$. Then the length of the line segment $P Q$ is equal to ________

Question

Aicharya · Accepted Answer

Line : $\frac{x+6}{-8}=\frac{y-7}{12}=\frac{z+5}{-16}$ $\begin{aligned} & \frac{\mathrm{x}+6}{2}=\frac{\mathrm{y}-7}{-3}=\frac{\mathrm{z}+5}{4}=\lambda \\ & \mathrm{Q}(2 \lambda-6,7-3 \lambda, 4 \lambda-5) \\ & \overline{\mathrm{QR}}(2 \lambda-7,-3 \lambda, 4 \lambda-11) \\ & \overline{\mathrm{QR}} \cdot \text { dr's of line }=0 \\ & 4 \lambda-14+9 \lambda+16 \lambda-44=0 \\ & 29 \lambda=58 \Rightarrow \lambda=2 \\ & \mathrm{Q}(-2,1,3) \\ & \mathrm{PQ}=\sqrt{144+9+16}=\sqrt{169}=13 \end{aligned}$

Let $P$ be the point $(10,-2,-1)$ and $Q$ be the foot of the perpendicular drawn from the point $R(1,7,6)$…

Solution