Question

How do you write `x^2 – x + y^2 + y = 0` in standard form and why type of conic is it?

Answer 1

Circle: `(x-1/2)^2+(y+1/2)^2=1/2`

Explanation:

Complete the square for both the `x` and `y` parts.

`color(blue)((x^2-x+?))+color(red)((y^2+y+?))=0`

`color(blue)((x^2-x+1/4))+color(red)((y^2+y+1/4))=1/4+1/4`

We add `1/4` to both the `x` and `y` portions because they will factor into `(x-1/2)^2` and `(y+1/2)^2`, respectively. Don't forget to add to the right side the exact same numbers you added to the left so the equation remains balanced.

`(x-1/2)^2+(y+1/2)^2=1/2`

This is a circle, since the standard form of a circle is:

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

There's a center at `(1/2,-1/2)` and a radius of `sqrt2/2`.