If the mirror image of the point (1, 3, 5) with respect to the plane
4x - 5y + 2z = 8 is (α, β, γ), then 5(α + β + γ) equals :

Solution

Maths sir · 4 steps

Walk from the point along the plane's normal, twice as far as the foot, and you land on the image.

1. Image formula. For a point (x₁, y₁, z₁) and the plane ax + by + cz + d = 0, the image lies on the normal through the point, at twice the distance to the foot. That gives one common ratio for all three coordinates.

   (α - x₁)/a = (β - y₁)/b = (γ - z₁)/c = -2(ax₁ + by₁ + cz₁ + d)/(a² + b² + c²)

2. Put in numbers. Write the plane as 4x - 5y + 2z - 8 = 0. At (1, 3, 5): 4 - 15 + 10 - 8 = -9, and a² + b² + c² = 16 + 25 + 4 = 45. So the common ratio is (-2 × (-9))/45 = 2/5.
3. Find the image. α = 1 + 4 × 2/5 = 13/5, β = 3 - 5 × 2/5 = 1, γ = 5 + 2 × 2/5 = 29/5. Check: the midpoint (9/5, 2, 27/5) gives 36/5 - 10 + 54/5 = 8, so it lies on the plane.
4. So. α + β + γ = (13 + 5 + 29)/5 = 47/5, so 5(α + β + γ) = 47, option (c).

The idea to keep. Image and foot use the same ratio formula: -2(…) on top for the image, -(…) for the foot. Keep the plane's constant on the left side before you substitute.

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