Question

How would you find the center and radius of `x^2 + y^2 - 6x=0`?

Answer 1

Center: `(3,0)`
Radius: `3`

Explanation:

The standard form for the equation of a circle with center `(color(red)a,color(blue)b)` and radius `color(green)r` is
`color(white)("XXX")(x-color(red)a)^2+(y-color(blue)b)^2=color(green)r^2`

We need to convert the given equation `x^2+y^2-6x=0`
into this standard form.

Grouping the terms containing `x`
`x^2-6xcolor(white)("xxx")+color(white)("xxx")y^2color(white)("xxx")=0`

The `(y-color(blue)b)^2` term in the standard form can be written directly as `(y-color(blue)0)^2`
but the writing the `(x-color(red)a)^2` will require some manipulation of `x^2-6x`; to do this we will "complete the square:
`x^2-6color(magenta)(+3^2)color(white)("x")+color(white)("x")(y-color(blue)0)^2color(white)("xx")=0color(magenta)(+3^2)`

`(x-color(red)3)^2color(white)("xxx")+color(white)("x")(y-color(blue)0)^2color(white)("x")=color(green)3^2`
`color(white)("XXX")`which is the required form for a circle with
`color(white)("XXX")` center at `(color(red)3,color(blue)0)` and
`color(white)("XXX")` radius `color(green)3`