Question

Let P be the plane passing through the point (1, 2, 3) and the line of intersection of the planes \(\vec{r} \cdot (\hat{i} + \hat{j} + 4\hat{k}) = 16\) and \(\vec{r} \cdot (-\hat{i} + \hat{j} + \hat{k}) = 6\). Then which of the following points does NOT lie on P ?

• A. (-8, 8, 6)
• B. (6, -6, 2)
• C. (4, 2, 2)
• D. (3, 3, 2)

Hint:

The concept of a family of planes (or lines, or circles) is a powerful technique. For planes, any plane passing through the intersection line of \(P_1=0\) and \(P_2=0\) can be written as \(P_1 + \lambda P_2 = 0\). A single additional condition (like passing through a point) is enough to determine the unique value of \(\lambda\).

Answer 1

The Correct Option is C

Step 1: Understanding the Question
We need to find the equation of a plane that belongs to the family of planes passing through the intersection of two given planes. We will use a given point to find the specific plane from this family. Then we will test which of the option points does not satisfy the plane's equation.
Step 2: Key Formula or Approach
The equation of any plane passing through the intersection of two planes \(P_1 = 0\) and \(P_2 = 0\) is given by the equation of the family of planes: \(P_1 + \lambda P_2 = 0\), where \(\lambda\) is a parameter.
Step 3: Detailed Explanation
Write the Cartesian equations of the given planes:
Plane 1 (\(P_1\)): \(x + y + 4z - 16 = 0\)
Plane 2 (\(P_2\)): \(-x + y + z - 6 = 0\)
Write the equation of the family of planes:
The equation of the required plane P is: \[ (x + y + 4z - 16) + \lambda(-x + y + z - 6) = 0 \] Find the value of \(\lambda\):
The plane P passes through the point (1, 2, 3). We substitute these coordinates into the equation: \[ (1 + 2 + 4(3) - 16) + \lambda(-1 + 2 + 3 - 6) = 0 \] \[ (15 - 16) + \lambda(4 - 6) = 0 \] \[ -1 + \lambda(-2) = 0 \] \[ -2\lambda = 1 \implies \lambda = -\frac{1}{2} \] Find the equation of plane P:
Substitute \(\lambda = -1/2\) back into the family equation: \[ (x + y + 4z - 16) - \frac{1}{2}(-x + y + z - 6) = 0 \] Multiply by 2 to clear the fraction: \[ 2(x + y + 4z - 16) - (-x + y + z - 6) = 0 \] \[ 2x + 2y + 8z - 32 + x - y - z + 6 = 0 \] \[ 3x + y + 7z - 26 = 0 \] Check which point does not lie on the plane:
We test each option by substituting its coordinates into the plane equation. (A) (-8, 8, 6): \(3(-8) + 8 + 7(6) - 26 = -24 + 8 + 42 - 26 = 50 - 50 = 0\). (Lies on P) (B) (6, -6, 2): \(3(6) + (-6) + 7(2) - 26 = 18 - 6 + 14 - 26 = 26 - 26 = 0\). (Lies on P) (C) (4, 2, 2): \(3(4) + 2 + 7(2) - 26 = 12 + 2 + 14 - 26 = 28 - 26 = 2 \neq 0\). (Does NOT lie on P) (D) (3, 3, 2): \(3(3) + 3 + 7(2) - 26 = 9 + 3 + 14 - 26 = 26 - 26 = 0\). (Lies on P) Step 4: Final Answer
The point (4, 2, 2) does not lie on the plane P.