Find the equation of the perpendicular bisector of the line joining P(2, -3) to Q(-5, 1)

To find the equation of the perpendicular bisector of the line segment joining $P(2, -3)$ and $Q(-5, 1)$, we need to follow two main steps: find the midpoint of the line segment and find the negative reciprocal of the slope of the line segment.

1. Find the Midpoint ($M$)

The perpendicular bisector passes through the midpoint of the line $PQ$. The formula for the midpoint is: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Using $P(2, -3)$ and $Q(-5, 1)$: $M = \left( \frac{2 + (-5)}{2}, \frac{-3 + 1}{2} \right) = \left( \frac{-3}{2}, \frac{-2}{2} \right) = \left( -1.5, -1 \right)$

2. Find the Slope of the Perpendicular Bisector ($m_\perp$)

First, find the slope of the line $PQ$ ($m_{PQ}$): $m_{PQ} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-3)}{-5 - 2} = \frac{4}{-7} = -\frac{4}{7}$

The slope of the perpendicular bisector ($m_\perp$) is the negative reciprocal of $m_{PQ}$: $m_\perp = -\frac{1}{m_{PQ}} = -\frac{1}{-4/7} = \frac{7}{4}$

3. Find the Equation of the Line

Using the point-slope form, $y - y_1 = m(x - x_1)$, where $(x_1, y_1) = (-1.5, -1)$ and $m = \frac{7}{4}$: $y - (-1) = \frac{7}{4}(x - (-1.5))$ $y + 1 = \frac{7}{4}(x + 1.5)$

Multiply the entire equation by 4 to clear the fraction: $4(y + 1) = 7(x + 1.5)$ $4y + 4 = 7x + 10.5$

To match the format of the options, multiply by 2 to remove the decimal: $8y + 8 = 14x + 21$

Rearrange the terms to one side: $8y - 14x + 8 - 21 = 0$ $8y - 14x - 13 = 0$

Comparing this result with the given options, it matches option C.

Exam Tip: Always remember that the product of the slopes of two perpendicular lines is $-1$ ($m_1 \times m_2 = -1$). This is a quick way to verify your slope calculation. Also, when dealing with fractions in coordinate geometry, multiplying through by the denominator early on often prevents arithmetic errors.