3D Geometry
Equal Angles with Axes — Sum of r Values
nta_pyq_2026_jan
Grade None
Question:
Let the line $L$ pass through the point $(-3,5,2)$ and make equal angles with the positive coordinate axes. If the distance of $L$ from the point $(-2,r,1)$ is $\sqrt{\dfrac{14}{3}}$, then the sum of all possible values of $r$ is:
Step-by-Step Solution
Key Concept: Line $L$ has direction $(1,1,1)$. $\vec{PQ}=(1,r-5,-1)$. $\vec{PQ}\times\hat{d}=(r-4,-2,6-r)$. Distance $=\sqrt{\tfrac{(r-4)^2+4+(6-r)^2}{3}}=\sqrt{\tfrac{14}{3}}$.
Sum of $r$ values $=10$.
Correct Answer: 1