To find an equation of the set of all points equidistant from points A(1, -5, 1) and B(3, 3, -5), we can follow these steps:
1. Find the midpoint of the line segment AB. The midpoint is the average of the x-coordinates, the average of the y-coordinates, and the average of the z-coordinates.
Midpoint coordinates: ((1+3)/2, (-5+3)/2, (1-5)/2) = (2, -1, -2)
2. Find the distance between the midpoint and one of the given points, A or B. Let's choose A.
Distance formula: √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
Distance between midpoint and A: √((1-2)^2 + (-5-(-1))^2 + (1-(-2))^2) = √(1 + 16 + 9) = √26
3. Write the equation using the distance found in step 2. Let (x, y, z) represent any point on the set of all points equidistant from A and B.
Equation: √((x-2)^2 + (y-(-1))^2 + (z-(-2))^2) = √26
Simplifying the equation, we get:
√((x-2)^2 + (y+1)^2 + (z+2)^2) = √26
This equation represents the set of all points that are equidistant from A(1, -5, 1) and B(3, 3, -5).
Hope this helps!