The plane through the intersection of the planes and and parallel to Y-axis also passes through the point
- A.(a)
- B.(b)
- C.(c)
- D.(d)
Show answer and solution
Correct option: C
Solution
The family of planes through the intersection is: (x + y + z - 1) + λ(2x + 3y - z + 4) = 0, giving (1+2λ)x + (1+3λ)y + (1-λ)z + (-1+4λ) = 0.
For the plane to be parallel to the Y-axis, its normal must be perpendicular to the Y-axis direction vector (0,1,0), so the coefficient of y must be zero: 1 + 3λ = 0 → λ = -1/3.
Substituting: (1 - 2/3)x + 0·y + (1 + 1/3)z + (-1 - 4/3) = 0 → (1/3)x + (4/3)z - (7/3) = 0 → x + 4z = 7.
Check (3, 2, 1): 3 + 4(1) = 7 ✓
For the plane to be parallel to the Y-axis, its normal must be perpendicular to the Y-axis direction vector (0,1,0), so the coefficient of y must be zero: 1 + 3λ = 0 → λ = -1/3.
Substituting: (1 - 2/3)x + 0·y + (1 + 1/3)z + (-1 - 4/3) = 0 → (1/3)x + (4/3)z - (7/3) = 0 → x + 4z = 7.
Check (3, 2, 1): 3 + 4(1) = 7 ✓
[JEE Main 2019 · 9 Jan · Shift I]