Question

The distance of the point \( (3, -5) \) from the line \( 3x - 4y - 26 = 0 \) is:

• A. \( \frac{3}{7} \)
• B. \( \frac{2}{5} \)
• C. \( \frac{7}{5} \)
• D. \( \frac{3}{5} \)
• E. \( 1 \)

Hint:

Always ensure the line equation is in the form \( Ax+By+C=0 \). In the denominator, the sum of squares \( 3^2 + 4^2 = 5^2 \) is a common Pythagorean triple, which simplifies calculations significantly.

Answer 1

The Correct Option is D

Concept: The perpendicular distance \( d \) from a point \( (x_1, y_1) \) to a line \( Ax + By + C = 0 \) is calculated using the formula: \[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \] This formula determines the shortest path between a specific coordinate and a straight line in a 2D plane.

Step 1: Identify coefficients and coordinates.
From the problem:
• Point \( (x_1, y_1) = (3, -5) \)
• Line coefficients: \( A = 3, B = -4, C = -26 \)

Step 2: Substitute into the distance formula.
\[ d = \frac{|3(3) + (-4)(-5) - 26|}{\sqrt{3^2 + (-4)^2}} \] \[ d = \frac{|9 + 20 - 26|}{\sqrt{9 + 16}} \] \[ d = \frac{|29 - 26|}{\sqrt{25}} \] \[ d = \frac{3}{5} \]