Question

The equation of a plane passing through the intersection of the planes \(x + 2y - 3z + 2 = 0\) and \(6x + y + z + 1 = 0\) and parallel to the line \(x - 1 = y + 2 = 7 - z\) is

• A. \(5x - y + 4z + 1 = 0\)
• B. \(5x + y + 4z + 1 = 0\)
• C. \(5x - y + 4z = 1\)
• D. \(5x + y + 4z = 1\)

Hint:

When a plane passes through the line of intersection of two planes, always start with their linear combination and then apply the given condition.

Answer 1

The Correct Option is C

Step 1: Write the family of planes.
A plane passing through the intersection of \[ x + 2y - 3z + 2 = 0 \quad \text{and} \quad 6x + y + z + 1 = 0 \] is given by \[ (x + 2y - 3z + 2) + \lambda(6x + y + z + 1) = 0 \] Step 2: Find the direction ratios of the given line.
From \(x - 1 = y + 2 = 7 - z\), let the common value be \(t\).
Then, the direction ratios of the line are \[ (1,\,1,\,-1) \] Step 3: Impose the parallel condition.
For the plane to be parallel to the given line, the direction ratios of the line must satisfy the plane equation.
Substituting into the plane family condition gives a suitable value of \(\lambda\).
Step 4: Obtain the required plane.
After simplification, the required plane equation is \[ 5x - y + 4z = 1 \]