The distance of the point $(1, 2, 3)$ from the plane $x + y - z = 5$ measured along the straight line $z = y = 2x$ is
Step-by-Step Solution
Key Concept: The equation of a plane through three points is found using the determinant method, and reflection of a point in a plane uses the perpendicular distance formula.
The plane passes through the points $(-2, -2, 2), (1, -1, 2), (1, 1, 1)$. Setting up the plane equation using the determinant method: $\begin{vmatrix} x + 2 & y + 2 & z - 2 \\ 3 & 1 & 0 \\ 3 & 3 & -1 \end{vmatrix} = 0$. Expanding this determinant yields $x - 3y - 6z = 0$. The image of the point $(-7, 2, 3)$ in this plane is found using reflection formulas, giving $(-1, 4, 4)$.
Correct Answer: 1