Question

A line passes through the point $A\left(2,3 , 5\right)$ and is parallel to the vector $2\hat{i}-\hat{j}+\hat{k}.$ If $P$ is a point on this line such that $AP=2\sqrt{6}$ , then the coordinates of point $P$ can be

• A. $\left(4,2 , 6\right)$
• B. $\left(6,1 , 7\right)$
• C. $\left(- 2,5 , - 3\right)$
• D. $\left(2,3 , 5\right)$

Answer 1

Answer: (B)

Equation of the line through $A\left(2,3 , 5\right)$ and parallel to $2\hat{i}-\hat{j}+\hat{k}$ is
$\frac{x - 2}{2}=\frac{y - 3}{- 1}=\frac{z - 5}{1}$
Coordinates of point $P$ on this line are $\left(2 \lambda + 2 , - \lambda + 3 , \lambda + 5\right)$
$AP=\sqrt{4 \lambda ^{2} + \lambda ^{2} + \lambda ^{2}}=2\sqrt{6}$
$\Rightarrow 6\lambda ^{2}=24\Rightarrow \lambda =\pm2$
Coordinates of point $P$ can be $\left(6,1 , 7\right)$ or $\left(- 2,5 , 3\right)$