3D Geometry
Points on Lines at Distance — Sum of Coordinates
nta_pyq_2026_jan
Grade 12
Question:
Let $L$ be the line $\dfrac{x+1}{2}=\dfrac{y+1}{3}=\dfrac{z+3}{6}$ and let $S$ be the set of all points $(a,b,c)$ on $L$, whose distance from the line $\dfrac{x+1}{2}=\dfrac{y+1}{3}=\dfrac{z-9}{0}$ along the line $L$ is 7. Then $\sum_{(a,b,c)\in S}(a+b+c)$ is equal to:
Step-by-Step Solution
Key Concept: Intersection $M$ of $L_1$ and $L_2$ (setting $z$-param of $L_2$=0): $M=(3,5,9)$... From solution $M=(3,5,9)$ is on $L$. $PM=7$ for points $P$ on $L$.
$\sum(a+b+c)=34$.
Correct Answer: 2