Question

How do use the method of translation of axes to sketch the curve `x^2+y^2+2x-6y+6=0`?

Answer 1

See the method below

Explanation:

We rewrite the equation by completing the squares

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

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

`(x+1)^2+(y-3)^2=4`

This is a circle, center `(-1,3)` and radius `=2`

We define new axes

`x'=x+1`

and `y'=y-3`

`(x')^2+(y')^2=4`

This is a circle, center `(0,0)` and radius `=2`

graph{((x+1)^2+(y-3)^2-4)(x^2+y^2-4)=0 [-10, 10, -5, 5]}