Question

Let P be the plane passing through the line \(\frac{x-1}{1}=\frac{y-2}{-3}=\frac{z+5}{7}\) and the point (2, 4, –3). If the image of the point (-1, 3, 4) in the plane P is (α, β, γ) then α + β + γ is equal to

• A. 9
• B. 10
• C. 11
• D. 12

Answer 1

The Correct Option is B

Given:
The equation of the plane is given by:
\[ \left|\begin{array}{ccc|c} x-1 & y-2 & z+5 & 0 \\ 1 & 2 & 2 & \\ 1 & -3 & 7 & \end{array}\right| = 0 \] 

The equation of the plane is:
\[ 4x - y - z = 7 \] 

Solving for:
\[ \frac{\alpha + 1}{4} = \frac{\beta - 3}{-1} = \frac{\gamma - 4}{-1} = \frac{-2(-4 - 3 - 4 - 7)}{16 + 1 + 1} = 2 \] 

Values of:
\[ \alpha = 7, \quad \beta = 1, \quad \gamma = 2 \] 

Finally, the sum:
\[ \alpha + \beta + \gamma = 10 \quad (\text{Option 2}) \]