Question

The distance between \( (2, 1, 0) \) and \( 2x + y + 2z + 5 = 0 \) is:

• A. \( 10 \)
• B. \( \frac{10}{3} \)
• C. \( \frac{10}{9} \)
• D. \( 5 \)
• E. \( 1 \)

Hint:

To ensure accuracy, always write the plane equation in standard form \( Ax+By+Cz+D=0 \) before identifying \( D \). If the constant was on the other side of the equals sign, its sign would change.

Answer 1

The Correct Option is B

Concept: The perpendicular distance \( d \) from a point \( (x_1, y_1, z_1) \) to a plane defined by the equation \( Ax + By + Cz + D = 0 \) is calculated using the following formula: \[ d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \] This formula determines the shortest path from a specific coordinate in 3D space to a flat surface.

Step 1: Identify the coordinates and plane coefficients.
From the problem, the point is \( (2, 1, 0) \), so:
• \( x_1 = 2 \)
• \( y_1 = 1 \)
• \( z_1 = 0 \) The equation of the plane is \( 2x + y + 2z + 5 = 0 \), so:
• \( A = 2 \)
• \( B = 1 \)
• \( C = 2 \)
• \( D = 5 \)

Step 2: Substitute values into the distance formula and solve.
\[ d = \frac{|2(2) + 1(1) + 2(0) + 5|}{\sqrt{2^2 + 1^2 + 2^2}} \] Calculate the numerator: \[ |4 + 1 + 0 + 5| = |10| = 10 \] Calculate the denominator: \[ \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \] Thus, the distance \( d = \frac{10}{3} \).