Let the plane P:r⃗ . a⃗ = d contain the line of intersection of two planes r⃗ . ( î + 3ĵ - k̂ ) = 6 and r⃗ . ( - 6î + 5ĵ - k̂ ) = 7. If the plane P passes through the point ( 2,3,1/2 ), then the value of (|13a⃗ |²)/d² is equal to :

Solution

Maths sir · 4 steps

Any plane through the line of intersection is P₁ + λ P₂ = 0. Use the point to fix λ.

1. Family of planes. In Cartesian form the two planes are x + 3y - z - 6 = 0 and -6x + 5y - z - 7 = 0. Every plane containing their line of intersection has this form:

   (x + 3y - z - 6) + λ(-6x + 5y - z - 7) = 0

2. Find lambda. Put in (2, 3, 1/2). The first bracket gives 2 + 9 - 0.5 - 6 = 4.5 and the second gives -12 + 15 - 0.5 - 7 = -4.5. So 4.5 - 4.5λ = 0, which gives λ = 1.
3. Normal and constant. With λ = 1 the plane is -5x + 8y - 2z = 13. So a⃗ = -5î + 8ĵ - 2k̂ and d = 13, which gives |a⃗|² = 25 + 64 + 4 = 93.
4. So. |13a⃗|²/d² = (169 × 93)/169 = 93, which is option (b). Scaling a⃗ and d by the same factor does not change this ratio, so any form of the plane's equation gives the same value.

The idea to keep. For a plane through the intersection of two planes, write P₁ + λ P₂ = 0 and let the given point decide λ. You never need the line itself.

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