Find the equation of the sphere if one of its diameters has endpoints (-4,-5,8) and (-1,1,17).

Answer 1

Recall that the center of the circle is directly in the middle of the any diameter and a radius (length from the center of the circle to any point on the edge of the circle) is exactly half of the diameter.

Note that the equation of a sphere is (x-h)²+(y-k)²+(z-l)² = r² where (h,k,l) is the center and r is the radius.

(a) Find the center of the circle: we simply need to find the midpoint between the two given points. We do this by finding the average of each of the coordinates. The mid point (and our center) is (-5/2, -2, 25/2)

(b) Find the radius: we find the distance between the two given points and divide by 2 to get the radius (distance between the center and a point on the edge of the circle). D² = (-4+1)² + (-5-1)² + (8-17)²

D² = 9 + 36 + 81 = 126 => D = √(126)

This then means that r = √(126)/2 => r² = 126/4 = 63/2

Putting both of these together gives us the equation (x+5/2)² + (y+2)² + (z-25/2)² = 63/2