Equation of the Locus of Points Equidistant from Two Fixed Points: An SEO-Optimized Guide

Understanding the equation of the locus of points that are equidistant from two fixed points is critical in geometry. This guide will walk you through the steps to derive the equation of such a locus, specifically for the points A(2, 3) and B(5, 7).

Introduction to the Locus of Points

The locus of points that are equidistant from two fixed points forms a special curve, which can be a circle, a line, or other geometric shapes, depending on the positions of the points. In this case, we are dealing with points in a two-dimensional plane, and the locus will be a straight line - specifically, the perpendicular bisector of the segment connecting the two points.

Steps to Find the Locus

Step 1: Determine the Midpoint of Segment AB

The first step is to find the midpoint of the segment AB. The coordinates of the midpoint M can be calculated as follows:

Coordinates of A: (2, 3)
Coordinates of B: (5, 7)

The formula for the midpoint is:

M (frac{x_1 x_2}{2}, frac{y_1 y_2}{2})

Plugging in the values:

M (frac{2 5}{2}, frac{3 7}{2}) (3.5, 5)

Step 2: Calculate the Slope of Line Segment AB

The next step is to find the slope of the line segment AB:

(text{slope of } AB frac{y_2 - y_1}{x_2 - x_1} frac{7 - 3}{5 - 2} frac{4}{3})

Step 3: Determine the Slope of the Perpendicular Bisector

The slope of the perpendicular bisector is the negative reciprocal of the slope of AB. Therefore:

(text{slope of perpendicular bisector} -frac{3}{4})

Step 4: Find the Equation of the Perpendicular Bisector

Using the point-slope form of the equation of a line, (y - y_1 m(x - x_1)), where (m) is the slope and ((x_1, y_1)) is a point on the line, we can substitute the slope and the midpoint (3.5, 5) into the equation:

(y - 5 -frac{3}{4}(x - 3.5))

Simplifying this, we get:

(y - 5 -frac{3}{4}x frac{10.5}{4})

(y - 5 -frac{3}{4}x 2.625)

(y -frac{3}{4}x 7.625)

Conclusion

Beneath the points A(2, 3) and B(5, 7), the locus of points equidistant from these two points is a straight line described by the equation:

(y -frac{3}{4}x 7.625)

This line is the perpendicular bisector of the segment AB, and it represents all the points that are equidistant from points A and B.

Keywords

locus of points, equidistant points, perpendicular bisector

Additional Resources

For a deeper understanding and practical applications of the concept, consider exploring resources on geometry, coordinate geometry, and algebraic equations of lines. Websites like Khan Academy, Google Scholar, and academic journals on geometry can provide ample material and exercises.

Graphs and visual representations can also be very helpful. Tools like GeoGebra or Desmos can be used to create interactive graphs and check the solutions derived here.