Question

How do you graph `x^2+y^2-4x+10y+20=0`?

Answer 1

Graphs a circle with centre `(2,-5)` and the radius `r=3`

Explanation:

This equation can be recognised as the equation of a circle - see Conic sections for a description of the patterns of the conic equations.

We need to reorganise and simplify it in order to get it into the standard form for a circle that will give us the centre and radius, after which it will be very easy to graph.

`x^2 -4x +y^2 +10y +20 =0`

Using completing the squares gives us
`(x-2)^2 -4 +(y+5)^2 -25 +20 = 0`

`(x-2)^2 +(y+5)^2 =9`

This is now in the form `(x-h)^2 + (y-k)^2 = r^2` where `(h,k)` is the centre and `r` is the radius.
therefore the centre is `(2,-5)` and the radius `r=3`

It is now possible to graph the circle.