Is this an equation of a circle?

Question

Wanted to know why the following equation doesn't represent a circle:

$2x^2 + 2y^2 − 6x + 4y + 7 = 0$

I know that $(\frac{-a}{2})^2 + (\frac{-b}{2})^2 - c \geq 0$

And it is, but the exercise says it doesn't represent a circle :/

Answer 1

If you complete squares you would get \begin{equation} \left(x-\frac{3}{4}\right)^2+\left(y+1\right)^2=-\frac{1}{4} \end{equation} It would mean that the circle has a radius equal to $r=\sqrt{-1/4}$, which is imaginary. So, it is not an equation of a circle.

Answer 2

Your equation is equivalent to:

$x^2+y^2-3 x+2y=-\frac{7}{2}$

Now add to both sides so as to complete the squares on the left hand side. Think about what you end up with on the right hand side.

Answer 3

Because this doesn't satisfy the g² +f² -c > 0 condition which an equation of the form x² +y² +2 gx +2 fy +c = 0 must to represent a circle.