Question

If the foot of perpendicular drawn from the point $\left(2,5 , 1\right)$ on a line passing through $\left(\alpha , 2 \alpha , 5\right)$ is $\left(\frac{1}{5} , \frac{2}{5} , \frac{3}{5}\right)$ , then $\alpha $ is equal to

• A. $\frac{19}{9}$
• B. $\frac{11}{570}$
• C. $\frac{57}{54}$
• D. $\frac{1}{25}$

Answer 1

Answer: (D)

From diagram $\overset{ \rightarrow }{P Q}\cdot \overset{ \rightarrow }{Q R}=0$
$\left[\left(2 - \frac{1}{5}\right) \hat{i} + \left(5 - \frac{2}{5}\right) \hat{j} + \left(1 - \frac{3}{5}\right) \hat{k}\right]\cdot \left[\left(\alpha - \frac{1}{5}\right) \hat{i} + \left(2 \alpha - \frac{2}{5}\right) \hat{j} + \left(5 - \frac{3}{5}\right) \hat{k}\right]=0$
$\Rightarrow \frac{9 \alpha }{5}-\frac{9}{25}+\frac{46 \alpha }{5}-\frac{46}{25}+\frac{44}{25}=0$
$\Rightarrow \alpha =\frac{1}{25}$