Question

Determine the center and radius of the following circle equation below!

Answer 1

The equation of a circle is given by the following general structure:

`(x-h)^2+(y-k)^2=r^2`

When solving the perfect squares we have:

`x^2-2hx+h^2+y^2-2ky+k^2=r^2^{}_{}`

When looking at the given expression, we can observe that -18x and 12y are the equivalents for -2hx and -2ky. We can use this information to find h and k which are the coordinates of the center of the circle:

`\begin{gathered} -2hx=-18x \\ h=(-18x)/(-2x) \\ h=9 \end{gathered}`
`\begin{gathered} -2ky=12y \\ k=(12y)/(-2y) \\ k=-6 \end{gathered}`

It means that the center of the circle is located at (9,-6).

To determine the radius of the circle, we have to take into account that 92 is the equivalent for h^2+k^2-r^2. Use this information to find r which is the radius of the circle:

`\begin{gathered} h^2+k^2-r^2=92 \\ 9^2+(-6)^2=92+r^2 \\ 81+36=92+r^2 \\ r^2=117-92 \\ r^2=25 \\ r=\sqrt[]{25} \\ r=5 \end{gathered}`

It means that the radius of the circle is 5.