Chapter 10.1, Problem 66E

To determine

To find: The center and radius of the sphere $x^2+y^2+z^2-10y=0$ .

Answer to Problem 66E

The center and radius of the sphere $x^2+y^2+z^2-10y=0$ are $\left(0,5,0\right)$ and $5$ respectively.

Explanation of Solution

Given information:

The equation of the sphere is $x^2+y^2+z^2-10y=0$ .

Calculation:

The general standard form of the equation of sphere is ${\left(x-h\right)}^2+{\left(y-k\right)}^2+{\left(z-j\right)}^2=r^2$ with center $\left(h,k,j\right)$ and $r$ radius.

Simplify the given equation of sphere.

  $\begin{array}{c}{x}^2+y^2+z^2-10y=0\\ x^2+\left(y^2-10y\right)+z^2=0\\ x^2+\left(y^2-10y+5^2-5^2\right)+z^2=0\\ x^2+{\left(y-5\right)}^2+z^2-25=0\\ {\left(x-0\right)}^2+{\left(y-5\right)}^2+{\left(z-0\right)}^2=5^2\end{array}$

Compare the equation of sphere ${\left(x-0\right)}^2+{\left(y-5\right)}^2+{\left(z-0\right)}^2=5^2$ with general equation. It gives the value of $h$ , $k$ , $j$ as $0$ , $5$ , $0$ respectively and the value of $r$ as $5$ .

Therefore, the center and radius of the sphere $x^2+y^2+z^2-10y=0$ are $\left(0,5,0\right)$ and $5$ respectively.