Let P be the plane passing through the point (1, 2, 3) and the line of intersection of the planes r⃗ . ( î + ĵ + 4k̂ ) = 16 and r⃗ . ( - î + ĵ + k̂ ) = 6. Then which of the following points does NOT lie on P?

Solution

Maths sir · 4 steps

Write the family of planes through the line, fix λ with (1, 2, 3), then test each point.

1. Family of planes. Any plane through the line of intersection has the form (x + y + 4z - 16) + λ(-x + y + z - 6) = 0.
2. Find λ. Put in (1, 2, 3). The first bracket gives 1 + 2 + 12 - 16 = -1 and the second gives -1 + 2 + 3 - 6 = -2. So -1 - 2λ = 0, which gives λ = -1/2.
3. Equation of P. Multiply through by 2: 2(x + y + 4z - 16) - (-x + y + z - 6) = 0. This simplifies to the plane below.

   3x + y + 7z = 26

   Point	3x + y + 7z	On P?
   (a) (3, 3, 2)	9 + 3 + 14 = 26	yes
   (b) (6, -6, 2)	18 - 6 + 14 = 26	yes
   (c) (4, 2, 2)	12 + 2 + 14 = 28	no
   (d) (-8, 8, 6)	-24 + 8 + 42 = 26	yes

4. So. Only (4, 2, 2) gives 28 instead of 26, so it is the point that does not lie on P. The answer is (c).

The idea to keep. For a plane through the intersection of two planes, write P₁ + λ P₂ = 0. One extra point fixes λ. Always check your final plane against that point before testing options.

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