Question

If the perpendicular bisector of the line segment joining \( P(1, 4) \) and \( Q(k, 3) \) has y-intercept -4, then the possible values of \( k \) are:

• A. -2 and 2
• B. -1 and 1
• C. -3 and 3
• D. -4 and 4

Hint:

Use the midpoint formula and slope relationships to find the equation of the perpendicular bisector.

Answer 1

The Correct Option is D

The equation of the perpendicular bisector of the line joining two points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) can be found by:
Step 1: Find the midpoint of \( P \) and \( Q \).
The midpoint \( M \) of the line segment joining \( P(1, 4) \) and \( Q(k, 3) \) is: \[ M = \left( \frac{1 + k}{2}, \frac{4 + 3}{2} \right) = \left( \frac{1 + k}{2}, \frac{7}{2} \right). \] Step 2: Find the slope of the line joining \( P \) and \( Q \).
The slope of the line joining \( P(1, 4) \) and \( Q(k, 3) \) is: \[ m = \frac{3 - 4}{k - 1} = \frac{-1}{k - 1}. \] Step 3: Find the slope of the perpendicular bisector.
The slope of the perpendicular bisector is the negative reciprocal of the slope of the line joining \( P \) and \( Q \): \[ \text{slope of the perpendicular bisector} = \frac{k - 1}{1}. \] Step 4: Use the point-slope form to write the equation of the perpendicular bisector.
The equation of the perpendicular bisector passing through the midpoint \( \left( \frac{1 + k}{2}, \frac{7}{2} \right) \) is: \[ y - \frac{7}{2} = \frac{k - 1}{1} \left( x - \frac{1 + k}{2} \right). \] Step 5: Use the y-intercept.
We are given that the y-intercept is -4. Setting \( x = 0 \), we substitute and solve for \( k \): \[ -4 - \frac{7}{2} = \frac{k - 1}{1} \left( 0 - \frac{1 + k}{2} \right). \] After solving this equation, the possible values of \( k \) are \( -4 \) and \( 4 \). Thus, the correct answer is: \[ \boxed{-4 \text{ and } 4}. \]