Question

If the planes \[ 2x - 5y + z = 8 \quad \text{and} \quad 2 \lambda x - 15y + 2z + 6 = 0 \] are parallel to each other, then the value of \( \lambda \) is

• A. \( \frac{1}{3} \)
• B. \( -3 \)
• C. \( 2 \)
• D. \( 3 \)

Hint:

For two planes to be parallel, their normal vectors must be proportional.

Answer 1

The Correct Option is A

Step 1: Condition for parallel planes.
For two planes to be parallel, their normal vectors must be parallel. The normal vector of a plane \( Ax + By + Cz = D \) is \( (A, B, C) \). Step 2: Find the normal vectors of the given planes.
For the plane \( 2x - 5y + z = 8 \), the normal vector is \( (2, -5, 1) \). For the plane \( 2 \lambda x - 15y + 2z + 6 = 0 \), the normal vector is \( (2 \lambda, -15, 2) \). Step 3: Apply the condition for parallelism.
The normal vectors must be proportional, so \[ \frac{2 \lambda}{2} = \frac{-15}{-5} = \frac{2}{1} \] Step 4: Solve for \( \lambda \).
Solving the equation \( \frac{2 \lambda}{2} = 3 \), we get \[ \lambda = \frac{1}{3} \] Step 5: Conclusion.
Thus, the value of \( \lambda \) is \( \frac{1}{3} \).