If the equation of the plane passing through the line of intersection of the planes 2x - 7y + 4z - 3 = 0, 3x - 5y + 4z + 11 = 0 and the point (-2, 1, 3) is ax + by + cz - 7 = 0, then the value of 2a + b + c - 7 is ____________.

Solution

Maths sir · 4 steps

Any plane through the intersection is P₁ + λ P₂ = 0; the given point fixes λ.

1. Family of planes. Every plane through the line of intersection of P₁ = 2x - 7y + 4z - 3 = 0 and P₂ = 3x - 5y + 4z + 11 = 0 can be written as P₁ + λ P₂ = 0 for some number λ.
2. Use the point. At (-2, 1, 3): P₁ = -4 - 7 + 12 - 3 = -2 and P₂ = -6 - 5 + 12 + 11 = 12. So -2 + 12λ = 0, which gives λ = 1/6.
3. Clear the fraction. Multiply P₁ + 1/6P₂ = 0 by 6 to get 6P₁ + P₂ = 0. The constant term becomes -18 + 11 = -7, which already matches the given form.

   15x - 47y + 28z - 7 = 0

4. So. a = 15, b = -47, c = 28, and 2a + b + c - 7 = 30 - 47 + 28 - 7 = 4.

The common slip: 11. Stopping at 2a + b + c and leaving out the final -7 gives 11, not 4.

The idea to keep. For a plane through the intersection of two planes, write P₁ + λ P₂ = 0 and let the extra condition find λ. Then scale the equation so its constant matches the form you are given.

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