Let Q be the mirror image of the point P(1, 0, 1) with respect to the plane S : x + y + z = 5. If a line L passing through (1, -1, -1), parallel to the line PQ meets the plane S at R, then QR² is equal to :

Solution

Maths sir · 4 steps

PQ is just the plane's normal (1, 1, 1), so find Q and R by walking along that direction.

1. Mirror image Q. At P, x + y + z - 5 = 1 + 0 + 1 - 5 = -3. Using the image formula with normal (1, 1, 1), (x-1)/1 = (y-0)/1 = (z-1)/1 = -2(-3)/3 = 2, so Q = (3, 2, 3).
2. Check Q. The midpoint of PQ is (2, 1, 2), and 2 + 1 + 2 = 5, so it lies on S. PQ has direction (2, 2, 2), which is along the normal (1, 1, 1).
3. Find R. L has direction (1, 1, 1), so a general point on it is (1 + t, -1 + t, -1 + t). Putting this point into S gives -1 + 3t = 5, so t = 2 and R = (3, 1, 1).

   QR² = (3-3)² + (2-1)² + (3-1)² = 0 + 1 + 4 = 5

4. So. QR² = 5, which is option (b).

The idea to keep. The line joining a point to its mirror image is always along the plane's normal. So any line parallel to PQ has the normal's direction ratios.

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