Introduction

Finding points that are equidistant from two given points is a classic problem in geometry and can be particularly useful in various fields such as engineering, design, and data analysis. This article will explore how to find the coordinates of the point on the x-axis that is equidistant from two given points, A and B. We'll go through step-by-step calculations to solve a specific example.

Problem Statement

Given the points A(-2, -3) and B(4, -1), we need to find the coordinates of the point on the x-axis that is equidistant from these two points. Let's denote this point as P(x, 0).

Solution: Using Distance Formulas

To find the point P on the x-axis that is equidistant from A and B, we need to set the distances from P to A and from P to B equal to each other. The general formula for the distance between two points (x1, y1) and (x2, y2) is:

D sqrt{(x2 - x1)^2 (y2 - y1)^2}

Let's apply this formula to find the distances from point P(x, 0) to points A(-2, -3) and B(4, -1).

Distance from P to A

Using the distance formula, the distance from P to A is:

DPA sqrt{(x - (-2))^2 ((0 - (-3))^2} sqrt{(x 2)^2 3^2}

Simplifying, we get:

DPA sqrt{x^2 4x 4 9} sqrt{x^2 4x 13}

Distance from P to B

Similarly, the distance from P to B is:

DPB sqrt{(x - 4)^2 ((0 - (-1))^2} sqrt{(x - 4)^2 1^2}

Simplifying, we get:

DPB sqrt{x^2 - 8x 16 1} sqrt{x^2 - 8x 17}

Solving for the Equidistant Point

Since the distances are equal, we set DPA equal to DPB:

sqrt{x^2 4x 13} sqrt{x^2 - 8x 17}

By squaring both sides, we eliminate the square roots and solve for x:

x^2 4x 13 x^2 - 8x 17

Combine like terms:

12x 4

x 1 / 3

Therefore, the coordinates of the point on the x-axis that is equidistant from points A and B are (1/3, 0).

Alternative Method: Perpendicular Bisector

Another way to approach this problem is by finding the midpoint of the line segment AB and the slope of the line passing through AB. The point we are looking for lies on the perpendicular bisector.

Midpoint Calculation

The midpoint M of the segment AB is calculated as:

M ((-2 4) / 2, (-3 (-1)) / 2) (1, -2)

Slope Calculation

The slope m_0 of the line AB is:

m_0 (-1 - (-3)) / (4 - (-2)) 2 / 6 1 / 3

The slope m_1 of the perpendicular bisector is the negative reciprocal of m_0:

m_1 -3

The equation of the perpendicular bisector through M(1, -2) and slope -3 is:

y - (-2) -3(x - 1)

y 2 -3x 3

y -3x 1

Intersection with yx

Since we are looking for the point on the x-axis, we set y x:

x -3x 1

4x 1

x 1 / 4

Thus, the coordinates of the point on the x-axis that is equidistant from points A and B are (1/4, 0).

Conclusion

In this article, we explored how to find the coordinates of the point on the x-axis that is equidistant from two given points using distance formulas and the concept of the perpendicular bisector. We specifically looked at points A(-2, -3) and B(4, -1) and found the coordinates (1/3, 0) and (1/4, 0).

Keywords: equidistant point, x-axis, midpoint