Three Dimensional Geometry Question 3
Question 3 - 2024 (01 Feb Shift 2)
Let $P$ and $Q$ be the points on the line $\frac{x+3}{8}=\frac{y-4}{2}=\frac{z+1}{2}$ which are at a distance of 6 units from the point $R(1,2,3)$. If the centroid of the triangle PQR is $(\alpha, \beta, \gamma)$, then $\alpha^{2}+\beta^{2}+\gamma^{2}$ is:
(1) 26
(2) 36
(3) 18
(4) 24
Show Answer
Answer (3)
Solution
$\mathrm{P}(8 \lambda-3,2 \lambda+4,2 \lambda-1)$
$\mathrm{PR}=6$
$(8 \lambda-4)^{2}+(2 \lambda+2)^{2}+(2 \lambda-4)^{2}=36$
$\lambda=0,1$
Hence $\mathrm{P}(-3,4,-1) & \mathrm{Q}(5,6,1)$
Centroid of $\triangle \mathrm{PQR}=(1,4,1) \equiv(\alpha, \beta, \gamma)$
$\alpha^{2}+\beta^{2}+\gamma^{2}=18$
