Question

How do you find the center and radius of the circle with equation `x^2 + y^2 + 10x +8y = 8`?

Answer 1

There are few ways which can be used to find the center and radius, one of the way is using completion of squares, and is given below in a step by step manner.

Explanation:

The given equation
`x^2+y^2+10x+8y = 8`

One way to do the problem is to use completing the square to get the problem in the form

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

Where `(h,k)` is the center and `r` is the radius.

`x^2+y^2+10x+8y=8`

Let us group the `x` and `y` together.

`x^2+10x + y^2 + 8y = 8`

Divide `x` coefficient by `2` square and add on both sides, do the same for `y` coefficient.

`x^2+10x+ (5)^2 + y^2 + 8y + (4)^2 = 8+5^2+4^2`
`(x+5)^2+(y+4)^2 = 8+25+16`
`(x+5)^2 + (y+4)^2 = 49`
`(x-(-5))^2 + (y-(-4))^2 = 7^2`

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

We can see the center `(h,k)` is `(-5,-4)`
Radius `r` is `7`