If the image of the point (4,4,3) in the line (x-1)/(2)=(y-2)/(1)=(z-1)/(3) is ( , , ), th…

Question

If the image of the point $(4,4,3)$ in the line $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-1}{3}$ is $(\alpha ,\beta ,\gamma )$, then $\alpha +\beta +\gamma $ is equal to

Accepted Answer

$$\begin{aligned}
&\begin{aligned}
& \overrightarrow{\mathrm{PQ}} \perp(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}}) \
& \Rightarrow 2(2 \lambda-3)+1(\lambda-2)+3(3 \lambda-2)=0 \
& \Rightarrow 14 \lambda-14=0, \lambda=1 \
& \text { So, } \mathrm{Q}(3,3,4)
\end{aligned}\
\&\text { Let image in } R(\alpha, \beta, \gamma)\
\&\begin{aligned}
& \frac{\alpha+4}{2}=3, \frac{\beta+4}{2}=3, \frac{\gamma+3}{2}=4 \
& (\alpha, \beta, \gamma)=(2,2,5) \
& \Rightarrow \alpha+\beta+\gamma=9
\end{aligned}
\end{aligned}$$

Solutions · 1