Question

The equation of the locus of points that are equidistant from the points \( (2,3) \) and \( (4,5) \) is:

• A. \( x + y = 0 \)
• B. \( x + y = 7 \)
• C. \( 4x + 4y = 38 \)
• D. \( x + y = 1 \)

Hint:

The perpendicular bisector of a line segment is the set of all points equidistant from the two given points.

Answer 1

The Correct Option is B

Step 1: Understanding the Perpendicular Bisector The locus of points equidistant from two given points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is the perpendicular bisector of the line segment joining these points.
Step 2: Finding the Midpoint of \( AB \) Given points: \[ A(2,3), \quad B(4,5) \] The midpoint of segment \( AB \) is: \[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{2+4}{2}, \frac{3+5}{2} \right) = \left( 3, 4 \right) \]
Step 3: Finding the Slope of \( AB \) \[ \text{Slope of } AB = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5-3}{4-2} = \frac{2}{2} = 1 \] Since the perpendicular bisector is perpendicular to \( AB \), its slope is: \[ m = -\frac{1}{1} = -1 \]
Step 4: Equation of the Perpendicular Bisector Using the point-slope formula: \[ y - y_0 = m(x - x_0) \] \[ y - 4 = -1(x - 3) \] \[ y - 4 = -x + 3 \] \[ x + y = 7 \]
Final Answer: \[ x + y = 7 \]