Finding the Equation of the Set of All Points Equidistant from Two Given Points

In coordinate geometry, the problem of finding the equation of the set of all points equidistant from two given points is an interesting and useful concept. This article will guide you through the step-by-step process of solving this problem.

Given Information

Let's consider two points in 3D space: A(-1, 6, 3) and B(5, 3, -3).

We need to find the equation of the set of all points that are equidistant from points A and B.

Step 1: Find the Midpoint of the Line Segment AB

The midpoint of the line segment AB is the point that is equidistant from both A and B. We can find the midpoint using the midpoint formula:

Midpoint = ((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2) Midpoint = (((-1) + 5)/2, (6 + 3)/2, (3 + (-3))/2) Midpoint = (2, 4.5, 0)

Step 2: Find the Equation of the Perpendicular Bisector

The set of all points equidistant from A and B lie on the perpendicular bisector of the line segment AB. To find the equation of the perpendicular bisector, we need to find a vector that is perpendicular to the line segment AB.

The vector AB is given by: AB = (5 - (-1), 3 - 6, -3 - 3) = (6, -3, 0)

A vector perpendicular to AB is: n = (a, b, c) = (-3, -6, 6)

The equation of the perpendicular bisector is: (x - 2) / (-3) = (y - 4.5) / (-6) = (z - 0) / 6

Simplifying, we get the equation of the perpendicular bisector: -3(x - 2) = -6(y - 4.5) = 6(z - 0) x - 2 = 2(y - 4.5) = -z

Conclusion

The equation of the set of all points equidistant from the points A(-1, 6, 3) and B(5, 3, -3) is: x - 2 = 2(y - 4.5) = -z

This equation represents the plane that passes through the midpoint (2, 4.5, 0) and is perpendicular to the line segment AB.