If the mirror image of the point (2, 4, 7) in the plane 3x - y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal to :

Solution

Maths sir · 4 steps

Walk from the point along the normal, twice the distance to the plane, and you land on the image.

1. Image formula. Write the plane as 3x - y + 4z - 2 = 0, so the normal is (3, -1, 4). The image of (x₁, y₁, z₁) is reached by moving along this normal, and the formula below gives how far.

   (a - 2)/3 = (b - 4)/-1 = (c - 7)/4 = -2(3(2) - 4 + 4(7) - 2)/(3² + (-1)² + 4²)

2. Put in numbers. The bracket is 6 - 4 + 28 - 2 = 28 and |n⃗|² = 9 + 1 + 16 = 26. So the common ratio is (-2 × 28)/26 = -28/13.
3. Find the image. a = 2 + 3(-28/13) = -58/13, b = 4 + 28/13 = 80/13, c = 7 + 4(-28/13) = -21/13.
4. So. 2a + b + 2c = (-116 + 80 - 42)/13 = -78/13 = -6, which is option (c).

The idea to keep. For the image in a plane ax + by + cz + d = 0, the common ratio is -2(ax₁ + by₁ + cz₁ + d)/(a² + b² + c²). For the foot of the perpendicular, use -1 instead of -2.

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