Let Q and R be the feet of perpendiculars from the point P(a, a, a) on the lines x=y, z=1 and x=-y, z=-1 respectively. If ∠ QPR is a right angle, then 12 a² is equal to _________.

Solution

Maths sir · 3 steps

Write each line in parametric form, find both feet, then set PQ⃗ · PR⃗ = 0.

1. Foot Q. Points on x = y, z = 1 are (t, t, 1), with direction (1, 1, 0). Then PQ⃗ = (t-a, t-a, 1-a), and making it perpendicular to the line gives 2(t-a) = 0, so t = a. This gives Q = (a, a, 1) and PQ⃗ = (0, 0, 1-a).
2. Foot R. Points on x = -y, z = -1 are (s, -s, -1), with direction (1, -1, 0). Then PR⃗ = (s-a, -s-a, -1-a), and the dot product with the direction is (s-a) - (-s-a) = 2s = 0, so s = 0. This gives R = (0, 0, -1) and PR⃗ = (-a, -a, -1-a).

   PQ⃗ · PR⃗ = 0 + 0 + (1-a)(-1-a) = a² - 1 = 0

3. So. a² = 1, so 12a² = 12. Taking a = -1 gives a genuine right angle: PQ⃗ = (0, 0, 2) and PR⃗ = (1, 1, 0). Taking a = 1 would put P on the first line, but a² is 1 either way.

The idea to keep. For a foot of perpendicular, take a general point on the line and make the joining vector perpendicular to the line's direction. A right angle at P means PQ⃗ · PR⃗ = 0.

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