3D Geometry
Distance from Point to Line — α²+β²
nta_pyq_2026_jan
Grade 12
Question:
If the distance of the point $P(43,\alpha,\beta)$, $\beta<0$, from the line $\vec{r}=4\hat{i}-\hat{k}+\mu(2\hat{i}+3\hat{k})$, $\mu\in\mathbb{R}$ along a line with direction ratios $3,-1,0$ is $13\sqrt{10}$, then $\alpha^2+\beta^2$ is equal to _____.
Step-by-Step Solution
Key Concept: Line passes through $A(4,0,-1)$, direction $\vec{d}=(2,0,3)$. $\vec{AP}=(39,\alpha,\beta+1)$. Cross product $\vec{AP}\times\vec{d}=(3\alpha,2\beta-115,-2\alpha)$... wait: $\vec{AP}\times\vec{d}=(0\cdot(\beta+1)-3\alpha, 3\cdot39-2(\beta+1), 0-0\cdot\alpha)$ — use full formula.
$\alpha^2+\beta^2=170$.
Correct Answer: 170