Let P( , , ) be the image of the point Q(1,6,4) in the line (x)/(1)=(y-1)/(2)=(z-2)/(3).

Question

Let $P(\alpha ,\beta ,\gamma )$ be the image of the point $Q(1,6,4)$ in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$. Then $2\alpha +\beta +\gamma $ is equal to \_\_\_\_\_\_\_\_

Answer: 11

Accepted Answer

$\begin{gathered} \overset{	o \leftarrow }{QR}\cdot (\hat{i}+2\hat{j}+3\hat{k})=0 \ (t-1)+(2t-5)	imes 2+(3t-2)	imes 3=0\Rightarrow t=\frac{17}{14} \ \Rightarrow R\equiv (\frac{17}{14},\frac{48}{14},\frac{79}{14}) \ \Rightarrow \frac{\alpha +1}{2}=\frac{17}{14},\frac{\beta +6}{2}=\frac{48}{14},\frac{\gamma +4}{2}=\frac{79}{14} \ 2\alpha +\beta +\gamma =\frac{68}{14}-2+\frac{96}{14}-6+\frac{158}{14}-4=11 \end{gathered}$

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