3D Geometry
Foot of Perpendicular — Length of Segment
nta_pyq_2024_apr
Grade 12
Question:
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 $PQ$ is equal to _____
Step-by-Step Solution
Key Concept: Direction of line through $(2,-5,11)$ and $(-6,7,-5)$: $(-8,12,-16)\propto(2,-3,4)$. Line: $\frac{x+6}{2}=\frac{y-7}{-3}=\frac{z+5}{4}=\lambda$. Point $Q=(2\lambda-6,-3\lambda+7,4\lambda-5)$. $\overrightarrow{QR}\perp(2,-3,4)$: find $\lambda$.
$Q=(-2,1,3)$. $PQ=13$.
Correct Answer: 13