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 $\left(2\lambda +2,-\lambda +3,\lambda +5\right)$
$AP=\sqrt{4{\lambda }^2+{\lambda }^2+{\lambda }^2}=2\sqrt{6}$
$\Rightarrow 6{\lambda }^2=24\Rightarrow \lambda =\pm 2$
Coordinates of point $P$ can be $\left(6,1,7\right)$ or $\left(-2,5,3\right)$