Question

If the line \(2x - 3y + 5 = 0\) is the perpendicular bisector of the line segment joining \(1, -2\) and \((a, b)\), then \(a + b =\)

• A. \(7\)
• B. \(1\)
• C. \(-1\)
• D. \(-7\)

Hint:

Remember that the perpendicular bisector of a line segment passes through the midpoint of the segment and is perpendicular to the segment.

Answer 1

The Correct Option is B

Step 1: Determine the midpoint of the line segment joining the points \( (1, -2) \) and \( (a, b) \). The coordinates of the midpoint are given by: \[ \left( \frac{1 + a}{2}, \frac{-2 + b}{2} \right). \] Step 2: Since \(2x - 3y + 5 = 0\) is the perpendicular bisector, the midpoint must satisfy this equation. Plugging in the midpoint coordinates, we get: \[ 2\left( \frac{1 + a}{2} \right) - 3\left( \frac{-2 + b}{2} \right) + 5 = 0. \] Simplifying the equation: \[ 1 + a - 3(-2 + b) + 5 = 0 \quad \Rightarrow \quad 1 + a + 6 - 3b + 5 = 0. \] This simplifies to: \[ a - 3b + 12 = 0. \] Step 3: Now, solve for \(a + b\). Rearranging the equation: \[ a - 3b = -12 \quad \Rightarrow \quad a + b = 1. \]