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

The problem asks us to find the equation of the locus of points equidistant from the points \( (2,3) \) and \( (4,5) \). To solve this, we must derive the perpendicular bisector of the line segment joining the two points. This bisector will be the locus of points equidistant from both points.

First, find the midpoint of the segment. The midpoint \( M \) is given by:
\( M = \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) \)
Substitute the given coordinates, \( (2,3) \) and \( (4,5) \):

\( M = \left(\frac{2+4}{2},\frac{3+5}{2}\right) = (3,4) \)

Next, find the slope of the line passing through \( (2,3) \) and \( (4,5) \):
\( \text{slope} = \frac{y_2-y_1}{x_2-x_1} = \frac{5-3}{4-2} = 1 \)

The slope of the perpendicular bisector is the negative reciprocal of this slope. Thus, the slope of the perpendicular bisector is:
\(-1\)

Using the point-slope form of a linear equation, \( y-y_1 = m(x-x_1) \), substitute \( m = -1 \) and the midpoint \( (3,4) \):
\( y-4 = -1(x-3) \)

Simplify:
\( y - 4 = -x + 3 \)
\( x + y = 7 \)

The equation of the locus of points equidistant from the points \( (2,3) \) and \( (4,5) \) is \( x + y = 7 \).

Answer 2

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: \[ x + y = 7 \]