Question

Let \(P(\alpha, \beta, \gamma)\) be the image of the point \(Q(1, 6, 4)\) in the line \[ \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}. \] Then \(2\alpha + \beta + \gamma\) is equal to _______.

Answer 1

Correct Answer: 11

Let the line be represented by:
\[x = t, \quad y = 2t + 1, \quad z = 3t + 2.\]
Given the point \(Q(1, 6, 4)\), we find the foot of the perpendicular \(A\) by letting:
\[A\left(\frac{17}{14}, \frac{48}{14}, \frac{79}{14}\right).\]
The direction vector of the line is:
\[\mathbf{b} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}.\]
To find the image point \(P(\alpha, \beta, \gamma)\), we reflect \(Q\) across \(A\) using:
\[\alpha = \frac{20}{14}, \quad \beta = \frac{12}{14}, \quad \gamma = \frac{102}{14}.\]
Calculating \(2\alpha + \beta + \gamma\):
\[2\alpha + \beta + \gamma = \frac{154}{14} = 11.\]
Answer: 11.