Question

Find the standard equation of the sphere. Endpoints of a diameter: \((1,0,0),(0,5,0)\)

Short answer

\((x - 0.5)^2 + (y - 2.5)^2 + z^2 = 26 / 4\)

Step-by-step solution

Finding the Center

The center of the sphere is the midpoint of the endpoints of the diameter. The midpoint between two points \((x_1,y_1,z_1)\) and \((x_2,y_2,z_2)\) can be found using the formula \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2})\). For the endpoints \((1,0,0)\) and \((0,5,0)\) of our diameter, this yields a center at \((\frac{1 + 0}{2}, \frac{0 + 5}{2}, \frac{0 + 0}{2}) = (0.5, 2.5, 0)\)

Finding the Radius

The radius of the sphere is half the distance between the endpoints of the diameter. The distance between the points \((x_1,y_1,z_1)\) and \((x_2,y_2,z_2)\) is given by \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\). For our diameter's endpoints, this gives a radius of \(\sqrt{(0 - 1)^2 + (5 - 0)^2 + (0 - 0)^2} = \sqrt{26}\). Half of this is \(\sqrt{26} / 2\).

Finding the Equation

Plugging the found center and radius values into the general equation of a sphere \((x - a)^2 + (y - b)^2 + (z - c)^2 = r^2\), the equation of the sphere is \((x - 0.5)^2 + (y - 2.5)^2 + z^2 = (\sqrt{26} / 2)^2\). Simplifying the radius on the right hand side, we get the sphere's equation to be \((x - 0.5)^2 + (y - 2.5)^2 + z^2 = 26 / 4\).