Question

The distance of the point \( (7, 5, 2) \) from the plane \( 3x + 4y + 2z - 8 = 0 \) measured parallel to the line \( \frac{x - 1}{3} = \frac{y - 2}{6} = \frac{z + 1}{2} \) is

• A. \( \sqrt{74} \) units
• B. \( \sqrt{47} \) units
• C. 6 units
• D. 7 units

Hint:

Use the distance formula with direction ratios to calculate the distance from a point to a plane along a given direction.

Answer 1

The Correct Option is D

Step 1: Find the direction ratios of the line.
The direction ratios of the line are \( \langle 3, 6, 2 \rangle \), derived from the given equation of the line.
Step 2: Use the formula for the distance from a point to a plane.
The distance \( D \) from a point \( (x_1, y_1, z_1) \) to a plane \( Ax + By + Cz + D = 0 \) measured along the direction ratios \( \langle a, b, c \rangle \) is given by: \[ D = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \] Substitute the point \( (7, 5, 2) \) and the plane equation \( 3x + 4y + 2z - 8 = 0 \) into the formula.
Step 3: Calculate the distance.
The distance from the point \( (7, 5, 2) \) to the plane along the direction ratios is 7 units.
Step 4: Conclusion.
Thus, the distance is 7 units.