Question

How do you find the center and radius for `x^2+2x+y^2+6y=6`?

Answer 1

Do a double completion of square, with respect to `x` and `y`, in order to convert to the arm `(x - a)^2 + (y - b)^2 = r`.

`1(x^2 + 2x) + 1(y^2 + 6y) = 6`

`1(x^2 + 2x + 1 - 1) + 1(y^2 + 6y + 9 - 9) = 6`

`1(x^2 + 2x + 1) - 1 + 1(y^2 + 6y + 9) - 9 = 6`

`(x + 1)^2 + (y + 3)^2 - 10 = 6`

`(x + 1)^2 + (y + 3)^2 = 16`

In the form `(x - a)^2 + (y - b)^2 = r`, the radius is given by `sqrt(r)` and the centre is at `(a, b)`. Then, the centre is at `(-1, -3)` and the radius measures `4` units.

Hopefully this helps!