Question

If \( (a,-6) \) lies on the perpendicular bisector of the line segment joining \( (-2,-1) \) and \( (4,-13) \), then the value of \( a \) is equal to

• A. \( 1 \)
• B. \( -2 \)
• C. \( 2 \)
• D. \( -3 \)
• E. \( 3 \)

Hint:

For a perpendicular bisector, first find the midpoint, then find the perpendicular slope.

Answer 1

Answer: (E)

The perpendicular bisector passes through the midpoint of the given points. Midpoint formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substituting \( (-2,-1) \) and \( (4,-13) \): \[ M = \left( \frac{-2 + 4}{2}, \frac{-1 + (-13)}{2} \right) \] \[ M = \left( \frac{2}{2}, \frac{-14}{2} \right) = (1, -7) \] The slope of the line joining these points is: \[ m = \frac{-13 + 1}{4 + 2} = \frac{-12}{6} = -2 \] The perpendicular bisector has the negative reciprocal slope: \[ m' = \frac{1}{2} \] Equation of the perpendicular bisector: \[ y - (-7) = \frac{1}{2} (x - 1) \] \[ y + 7 = \frac{1}{2} x - \frac{1}{2} \] \[ y = \frac{1}{2} x - \frac{1}{2} - 7 \] \[ y = \frac{1}{2} x - \frac{15}{2} \] Substituting \( (-6, a) \): \[ -6 = \frac{1}{2} a - \frac{15}{2} \] Multiplying by 2: \[ -12 = a - 15 \] \[ a = 3 \] 
Final Answer: \[ \boxed{3} \]