Condition of conic to be a circle

Question

Let $a,b,h,g,f,c \in \mathbb{R}$. Then the general equation of a conic given by: $ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0$ represents the equation of a circle iff \begin{align} \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} \leq 0, h = 0, a = b \neq 0. \end{align} For the $\Leftarrow$, it is easy to see as if you plug in the values, it can be easily written as $(x + \frac{g}{a})^2 + (y + \frac{f}{a})^2 = \frac{1}{a^2}(g^2 + f^2 - ac) $ which is a circle. I am having trouble with the $\Rightarrow$ part.

I am using the following definition of a circle:

A circle is a set of points $(x,y) \in \mathbb{R}^2$ which satisfies the equation: $(x- X)^2 + (y - Y)^2 = R^2$ for some $(X,Y) \in \mathbb{R}^2$, $R \in \mathbb{R}_{\geq 0}$.

Using this definition of a circle, how do I show the $\Rightarrow$ part of the above result for the general conic to be a circle?

Answer 1

@amd: Since my comment was too big, I had to post it as an answer.

$S \subseteq \mathbb{R}^2$ is defined as a circle if $\exists X,Y \in \mathbb{R}$, $R \in \mathbb{R}_{\geq 0}$ such that $(x - X)^2 + (y - Y)^2 = R^2 \ \ \forall (x,y) \in S$.

'$\Leftarrow$' proof:

If it is given $ a = b \neq 0$, $h = 0$, $\begin{pmatrix} a & h & g \\ h & b & f \\ g & f & c \end{pmatrix} \leq 0$ then $ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0$ can be rewritten as $(x + \frac{g}{a})^2 + (y + \frac{f}{b})^2 = \frac{1}{a^2}(g^2 + f^2 - ac)$. So I can give (X,Y) = $(-\frac{g}{a},-\frac{f}{b})$ and $R = \frac{1}{a}\sqrt{g^2 + f^2 - ac} \geq 0$ such that every $(x,y)$ which satisfies the locus $ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0$ satisfies $(x-X)^2 + (y-Y)^2 = R^2 $. Hence the set of points which satisfies $ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0$ is a circle.

'$\Rightarrow$' proof:

It is given that $ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0$ represents a circle. That is $\exists X,Y \in \mathbb{R}$, $R \in \mathbb{R}_{\geq 0}$ such that $(x - X)^2 + (y - Y)^2 = R^2$ for all $(x,y) $ satisfying the equation $ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0$. Now how do I proceed from here?

If I am supposed to match the terms, what is the formal reasoning to do so? Agreed that $x^2,y^2 ... $ etc are LI terms, and then how to go about from there?