Question

Let the image of the point P(1, 2, 6) in the plane passing through the points A(1, 2, 0), B(1, 4, 1) and C(0, 5, 1) be Q (α, β, γ). Then (α² + β² + γ²) is equal to

• A. 62
• B. 65
• C. 70
• D. 76

Answer 1

The Correct Option is B

Step 1: The equation of the plane passing through points A(1, 2, 0), B(1, 4, 1), and C(0, 5, 1) is: \[ A(x - 1) + B(y - 2) + C(z - 0) = 0 \] Using the coordinates of points A, B, and C, we get the system of equations:
From point \( (1, 4, 1) \), we get: \( 2B + C = 0 \)
From point \( (0, 5, 1) \), we get: \( -A + 3B + C = 0 \)
Solving this system, we get \( A = -2B \) and \( C = -2B \).
Step 2: Using the formula for the image of the point \( P(1, 2, 6) \):
\[ \frac{\alpha - 1}{1} = \frac{\beta - 2}{1} = \frac{\gamma - 6}{-2} = \frac{-2(1 + 2 - 12 - 3)}{6} \] Solving this, we get: \[ \alpha = 5, \beta = 6, \gamma = -2 \] 
Step 3: Now, calculate \( \alpha^2 + \beta^2 + \gamma^2 \): \[ \alpha^2 + \beta^2 + \gamma^2 = 5^2 + 6^2 + (-2)^2 = 25 + 36 + 4 = 65 \]