Question

The equation of the plane parallel to the plane \( 3x - 5y + 4z = 11 \) is:

• A. \( 3x - 5y + 4z = 21 \)
• B. \( 3x + 5y + 4z = 25 \)
• C. \( 3x + 5y + 4z = 35 \)
• D. none of these

Hint:

For two planes to be parallel, they must have the same normal vector, meaning the coefficients of \( x, y, z \) should be the same, but the constant term \( D \) must be different.

Answer 1

The Correct Option is D

The equation of a plane is given by the general form: \[ Ax + By + Cz = D \] where \( A, B, C \) are the coefficients of the plane and \( D \) is a constant. The key point here is that parallel planes have the same normal vector, which means their coefficients \( A, B, C \) are identical, but their constant term \( D \) is different. The given plane is: \[ 3x - 5y + 4z = 11 \] A plane parallel to this one will have the same coefficients of \( x, y, z \), but the constant term will be different. Therefore, the equation of a parallel plane will have the form: \[ 3x - 5y + 4z = k \] where \( k \) is any constant different from 11. Thus, the correct answer is: \[ \boxed{\text{(D) none of these}} \]