Question

The square of the distance of the image of the point \( (6, 1, 5) \) in the line \[ \frac{x - 1}{3} = \frac{y}{2} = \frac{z - 2}{4}, \] from the origin is _____ .

Answer 1

Correct Answer: 62

Let \( M(3\lambda + 1, 2\lambda, 4\lambda + 2) \)

\(\overrightarrow{AM} \cdot \mathbf{b} = 0\)
\(\implies 9\lambda - 15 + 4\lambda - 2 + 16\lambda - 12 = 0\)
\(\implies 29\lambda = 29  \implies \lambda = 1\)

So,
\(M(4, 2, 6), \quad I = (2, 3, 7)\)

Hence Distance =
\(\sqrt{4 + 9 + 49} = \sqrt{62}\)

Therefore, the square of the distance of the image of the point is 62.