Let a plane P pass through the point (3, 7, -7) and contain the line, (x - 2)/(- 3) = (y - 3)/2 = (z + 2)/1. If distance of the plane P from the origin is d, then d² is equal to ______________.

Solution

Maths sir · 4 steps

Take two vectors lying in the plane and cross them; that gives the normal, and the plane follows.

1. Two vectors in P. The line passes through B(2, 3, -2) with direction (-3, 2, 1), and both lie in P. The given point A(3, 7, -7) is also in P, so AB⃗ = (-1, -4, 5) is a second vector in P.

   n⃗ = |î, ĵ, k̂; -1, -4, 5; -3, 2, 1| = -14î - 14ĵ - 14k̂

2. Equation of P. The normal is along (1, 1, 1), so P is x + y + z = c. Putting in B gives c = 2 + 3 - 2 = 3. Check with A: 3 + 7 - 7 = 3.
3. Distance from origin. d = (|0 + 0 + 0 - 3|)/√(1² + 1² + 1²) = 3/√3 = √3.
4. So. d² = 3, so the answer is 3.

The common slip: 9. Taking the distance as just |c| = 3 without dividing by |n⃗| = √3 gives d² = 9, not 3.

The idea to keep. A plane through a point and a line: join the point to any point on the line, cross that vector with the line's direction, and you have the normal.

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