Question

How do you find the center (h,k) and radius r of the circle with the given equation `x^2+y^2-18x+18y=-137`?

Answer 1

The centre is `(9,-9)` and the radius is `5`

Explanation:

The equation needs to be rearranged into the form `(x-h)^2 + (y-k)^2 = r^2`

First reorder the expression to group the `x` and `y` terms
`x^2 - 18x +y^2 +18y = -137`

Each group then needs to be expressed as a square by completing the squares

`(x-9)^2 -9^2 + (y+9)^2 -9^2 = -137`
`(x-9)^2 +(y+9)^2 = -137 +81 +81 = 25`

The circle is then `(x-9)^2 + (y+9)^2 = 25`

The centre is therefore `(9,-9)` and the radius is `5`