Let the mirror image of the point (a, b, c) with respect to the plane 3x - 4y + 12z + 19 = 0 be (a - 6, β, γ). If a + b + c = 5, then 7β - 9γ is equal to ______________.

Solution

Maths sir · 5 steps

The x-coordinate drops by 6, so the image formula tells you the step size along the normal at once.

1. Image formula. The normal to the plane is (3, -4, 12), and 3² + 4² + 12² = 169. The image (x, y, z) of (a, b, c) satisfies the relation below.

   (x-a)/3 = (y-b)/-4 = (z-c)/12 = (-2 (3a-4b+12c+19))/169

2. Use the x-shift. Here x - a = -6, so the common ratio is -6/3 = -2. That gives y - b = (-4)(-2) = 8 and z - c = 12 × (-2) = -24, so β = b + 8 and γ = c - 24.
3. Plane condition. Setting the right side equal to -2 gives 3a - 4b + 12c + 19 = 169, so 3a - 4b + 12c = 150. Put a = 5 - b - c: 15 - 7b + 9c = 150, so 7b - 9c = -135.
4. Combine. 7β - 9γ = 7(b + 8) - 9(c - 24) = (7b - 9c) + 56 + 216 = -135 + 272.
5. So. 7β - 9γ = 137.

The common slip: -32. Using the foot-of-perpendicular formula (no factor 2) gives 3a - 4b + 12c = 319 and an answer of -32, not 137.

The idea to keep. For a mirror image, all three coordinate shifts are the same multiple of the normal. One known shift fixes that multiple, and the multiple fixes the plane condition on the point.

Answer from the NTA final answer key. Solution by PrepWiser.