Write an equation of a sphere if one of its diameters has endpoints $(-5,0,3)$ and $(-11,-4,5)$.
Solution
Step 1 ：Given the endpoints of the diameter of the sphere as $(-5,0,3)$ and $(-11,-4,5)$, we can find the center of the sphere by calculating the midpoint of these two points. The midpoint of two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is given by $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)$. Substituting the given points, we find the center of the sphere to be $(-8.0, -2.0, 4.0)$.
Step 2 ：Next, we calculate the radius of the sphere. The distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is given by $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$. The radius is half of this distance. Substituting the given points, we find the radius to be approximately $3.7416573867739413$.
Step 3 ：The equation of a sphere is given by $(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2$, where $(a,b,c)$ is the center of the sphere and $r$ is the radius. Substituting the values we found for the center and the radius, we get the equation of the sphere as $(x - 8.0)^2 + (y - 2.0)^2 + (z - -4.0)^2 = 14.0$.
Step 4 ：Final Answer: The equation of the sphere is \(\boxed{(x - 8.0)^2 + (y - 2.0)^2 + (z - -4.0)^2 = 14.0}\).
