Question

Find the equation of the sphere passing through pleft parenthesis negative 4 comma 7 comma 6 right parenthesis(−4, 7, 6) and qleft parenthesis 6 comma negative 5 comma 7 right parenthesis(6, −5, 7) with its center at the midpoint of pq.

Answer 1

The equation of a sphere is:

(x – h)^2 + (y – k)^2 + (z – l)^2 = r^2

where h, k and l are the coordinates of the center of the sphere

 

Using the midpoint formula, the coordinate of the center is:

h = (-4 + 6) / 2 = 1

k = (7 + -5) / 2 = 1

l = (6 + 7) / 2 = 6.5

 

so the equation becomes:

(x – 1)^2 + (y – 1)^2 + (z - 6.5)^2 = r^2

 

we plug in any point, in this case point P to solve for r:

(-4 -1)^2 + (7 – 1)^2 + (6 - 6.5)^2 = r^2

r^2 = 61.25

 

So the full equation is:

(x – 1)^2 + (y – 1)^2 + (z - 6.5)^2 = 61.25