Question

If the points \( (1, 1, \lambda) \) and \( (-3, 0, 1) \) are equidistant from the plane \( 3x + 4y - 12z + 13 = 0 \), then the integer value of \( \lambda \) is

• A. 2
• B. 1
• C. 3
• D. 4

Hint:

To find the distance from a point to a plane, use the distance formula and set the distances equal when points are equidistant.

Answer 1

The Correct Option is B

Step 1: Use the distance formula.
The distance from a point \( (x_1, y_1, z_1) \) to the plane \( Ax + By + Cz + D = 0 \) is given by: \[ \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \] We apply this formula for both points \( (1, 1, \lambda) \) and \( (-3, 0, 1) \).
Step 2: Set the distances equal.
Since the points are equidistant, we equate the distances from the plane for the two points. After solving for \( \lambda \), we find that \( \lambda = 1 \).
Step 3: Conclusion.
The correct answer is (B) 1.