Question

Let a line $L$ pass through the point $P(2,3,1)$ and be parallel to the line $x+3 y-2 z-2=0=x-y+2 z$ If the distance of $L$ from the point $(5,3,8)$ is $\alpha$, then $3 \alpha^2$ is equal to_______

Answer 1

∣∣​i^11​j^​3−1​k^−22​∣∣​=4i^−4j^​−4k^
∴ Equation of line is 1x−2​=−1y−3​=−1z−1​
Let Q be (5,3,8) and foot of ⊥ from Q on this line be R.
Now, R≡(k+2,−k+3,−k+1)
DR of QR are (k−3,−k,−k−7)
∴(1)(k−3)+(−1)(−k)+(−1)(−k−7)=0
⇒k=−34​
∴α2=(313​)2+(34​)2+(317​)2=9474​
∴3α2=158
So, the correct answer is 158.