Question

The line \( \frac{x - 2}{3} = \frac{y - 3}{4} = \frac{z - 4}{5} \) is parallel to the plane:

• A. \( 3x + 4y + 5z = 7 \)
• B. \( 2x + 3y + 4z = 0 \)
• C. \( x + y - z = 2 \)
• D. \( 2x + y - 2z = 0 \)

Hint:

To find the equation of a plane parallel to a line, use the direction ratios of the line as the coefficients of the plane equation.

Answer 1

The Correct Option is D

The general equation for a line in parametric form is given by: \[ \frac{x - 2}{3} = \frac{y - 3}{4} = \frac{z - 4}{5} \] Let \( t \) be the parameter, so the parametric equations of the line become: \[ x = 3t + 2, \quad y = 4t + 3, \quad z = 5t + 4 \] We are given that the line is parallel to a plane. The direction ratios of the line are \( \langle 3, 4, 5 \rangle \), and the normal to the plane will have the same direction ratios. Therefore, the equation of the plane can be written as: \[ 2x + y - 2z = 0 \] Thus, the correct answer is \( 2x + y - 2z = 0 \).