Question

Let the image of the point $P (1,2,3)$ in the line $L : \frac{ x -6}{3}=\frac{ y -1}{2}=\frac{ z -2}{3}$ be $Q .$ let $R (\alpha, \beta, \gamma)$ be a point that divides internally the line segment $PQ$ in the ratio $1: 3$. Then the value of $22(\alpha+\beta+\gamma)$ is equal to

Answer 1

Let $M$ be the mid-point of PQ
$\therefore M =(3 \lambda+6,2 \lambda+1,3 \lambda+2)$
Now, $\overrightarrow{ PM }=(3 \lambda+5) \hat{ i }+(2 \lambda-1) \hat{ j }+(3 \lambda-1) \hat{ k }$
$\because \overrightarrow{ PM } \perp(3 \hat{ i }+2 \hat{ j }+3 \hat{ k }) $
$\therefore 3(3 \lambda+5)+2(2 \lambda-1)+3(3 \lambda-1)=0$
$\lambda=\frac{-5}{11}$
$\therefore M \left(\frac{51}{11}, \frac{1}{11}, \frac{7}{11}\right)$
Since R is mid-point of PM
$22(\alpha+\beta+\gamma)=125$