4.3 Q-25

Question Statement

Find the equation of a line through the point (5, -8) that is perpendicular to the line joining points A(-15, -8) and B(10, 7).

Background and Explanation

To solve this problem, we need to use the concept of perpendicular lines. Two lines are perpendicular if the product of their slopes is -1. First, we calculate the slope of the line joining points A and B, and then use the fact that the slope of the perpendicular line is the negative reciprocal of the original slope.

Solution

Step 1: Find the slope of the line AB.

The slope of a line passing through two points $(x_1, y_1)$ and ( $x_2, y_2$ ) is given by the formula: $\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute the coordinates of points A(-15, -8) and B(10, 7) into this formula: $\text{slope of AB} = \frac{7 - (-8)}{10 - (-15)} = \frac{7 + 8}{10 + 15} = \frac{15}{25} = \frac{3}{5}$

Step 2: Find the slope of the perpendicular line.

The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Since the slope of AB is $\frac{3}{5}$ , the slope of the perpendicular line will be: $m_{\text{perpendicular}} = \frac{-5}{3}$

Step 3: Use the point-slope form of the equation of a line.

The point-slope form is given by: $y - y_1 = m(x - x_1)$

Here, $m = \frac{-5}{3}$ is the slope of the required line, and the point (5, -8) is the point through which the line passes. Substituting these values: $y - (-8) = \frac{-5}{3}(x - 5)$

Step 4: Simplify the equation.

Simplify both sides: $y + 8 = \frac{-5}{3}(x - 5)$

Multiply through by 3 to eliminate the denominator: $3(y + 8) = -5(x - 5)$

Distribute both sides: $3y + 24 = -5x + 25$

Rearrange the equation: $5x + 3y - 1 = 0$

Thus, the equation of the required line is: $5x + 3y - 1 = 0$

Key Formulas or Methods Used

Slope Formula: The slope of a line through two points is given by $\frac{y_2 - y_1}{x_2 - x_1}$ .
Perpendicular Line Slope: The slope of a perpendicular line is the negative reciprocal of the original line’s slope, i.e., $m_{\text{perpendicular}} = \frac{-1}{m_{\text{original}}}$ .
Point-Slope Form: $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Summary of Steps

Find the slope of the line joining points A(-15, -8) and B(10, 7) using the slope formula.
Take the negative reciprocal of this slope to find the slope of the perpendicular line.
Use the point-slope form of the equation of a line to set up the equation with the slope $\frac{-5}{3}$ and the point (5, -8).
Simplify the equation to obtain the final answer: $5x + 3y - 1 = 0$ .