Find the coordinates of points on line $\frac{x}{1} = \frac{y-1}{2} = \frac{z+1}{2}$ which are at a distance of $\sqrt{11}$ units from origin.

$(1, 3, 1)$ and $(-1, -1, -3)$

The correct answer is Option (1) → $(1, 3, 1)$ and $(-1, -1, -3)$ ##

Given line is, $\frac{x}{1} = \frac{y-1}{2} = \frac{z+1}{2} = \lambda \text{ (say) } \lambda \in \mathbb{R}$

Any point on this line is of the form $P(\lambda, 2\lambda+1, 2\lambda-1)$

Distance of this point from origin = $\sqrt{11}$

$\sqrt{(\lambda)^2 + (2\lambda+1)^2 + (2\lambda-1)^2} = \sqrt{11}$

$\lambda^2 + 4\lambda^2 + 1 + 4\lambda + 4\lambda^2 + 1 - 4\lambda = 11$

$9\lambda^2 = 9 \Rightarrow \lambda = \pm 1$

CASE 1: If $\lambda = 1$, Coordinate of reqd. point $= (1, 3, 1)$

CASE 2: If $\lambda = -1$, Coordinate of reqd. point $= (-1, -1, -3)$