Question

The equation of a plane passing through the line of intersection of the planes \( x + 2y + 3z = 2 \) and \( x - y + z = 3 \) and at a distance \( \frac{2}{\sqrt{3}} \) from the point \( (3, 1, -1) \) is:

• A. \( 5x - 11y + z = 17 \)
• B. \( \sqrt{2}x + y = 3\sqrt{2} - 1 \)
• C. \( x + y + z = \sqrt{3} \)
• D. \( x - \sqrt{2}y = 1 - \sqrt{2} \)

Hint:

The equation of the plane through the intersection of two planes can be found by combining their equations with a parameter and using the distance formula.

Answer 1

The Correct Option is A

The equation of the plane passing through the line of intersection of two planes \( P_1 \) and \( P_2 \) is: \[ P_1 + \lambda P_2 = 0 \] where \( \lambda \) is a constant. Substituting the equations of the planes \( P_1: x + 2y + 3z = 2 \) and \( P_2: x - y + z = 3 \), we get the equation: \[ (x + 2y + 3z) + \lambda(x - y + z) = 0 \] Simplifying: \[ (1 + \lambda)x + (2 - \lambda)y + (3 + \lambda)z = 2 + 3\lambda \] Now, substitute the given distance from the point \( (3, 1, -1) \) to the plane and solve for \( \lambda \), which will give us the equation \( 5x - 11y + z = 17 \).