Question

The equation of a plane passing through \( (-1, 2, 3) \) and whose normal makes equal angles with the coordinate axes is:

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

Hint:

When the normal vector makes equal angles with the axes, the coefficients in the equation of the plane are equal.

Answer 1

The Correct Option is C

Step 1: General form of the plane equation.
The general equation of a plane is: \[ a(x - x_1) + b(y - y_1) + c(z - z_1) = 0 \] where \( (x_1, y_1, z_1) \) is a point on the plane, and \( a, b, c \) are the components of the normal vector to the plane. We are given that the plane passes through \( (-1, 2, 3) \). Therefore, the equation becomes: \[ a(x + 1) + b(y - 2) + c(z - 3) = 0 \]
Step 2: Condition for equal angles.
The normal vector to the plane makes equal angles with the coordinate axes. This implies that the components of the normal vector, \( a, b, c \), are all equal, i.e., \( a = b = c \).
Step 3: Substitute and simplify.
Substituting \( a = b = c \) in the equation: \[ a(x + 1) + a(y - 2) + a(z - 3) = 0 \] Simplifying: \[ a(x + y + z - 4) = 0 \] Since \( a \neq 0 \), we get: \[ x + y + z - 4 = 0 \]