43. Under what conditions does the general second-order equation:

$$
a_{11} x^{2}+2 a_{12} x y+a_{22} y^{2}+2 a_{13} x+2 a_{23} y+a_{33}=0
$$

represent a circle on the coordinate plane?
43. If we expand the parentheses in the canonical equation of a circle, we get:

$$
x^{2}-2 a x+a^{2}+y^{2}-2 b y+b^{2}-R^{2}=0
$$

Compared to the general second-order equation, this equation has two peculiarities: 1) the coefficients of $x^{2}$ and $y^{2}$ are equal; 2) there is no term containing $x y$. Consider a general equation with these properties $k x^{2}+k y^{2}+2 m x+2 n y+p=0$ and try to bring it to the equation of a circle. Dividing both parts by $k$, we get:

$$
x^{2}+y^{2}+2 \frac{m}{k} x+2 \frac{n}{k} y+\frac{p}{k}=0
$$

Let $\frac{m}{k}=-a, \frac{n}{k}=-b$ and add and subtract $a^{2}$ and $b^{2}$; we get: $(x-a)^{2}+(y-b)^{2}+\frac{p}{k}-a^{2}-b^{2}=0$. In the case when $a^{2}+b^{2}-\frac{p}{k}>0$, we can set $a^{2}+b^{2}-\frac{p}{k}=R^{2}$, and then we obtain the equation of a circle: $(x-a)^{2}+(y-b)^{2}=R^{2}$. If $a^{2}+b^{2}-\frac{p}{k}<0$, then no point $(x ; y)$ satisfies our equation. The case $a^{2}+b^{2}-\frac{p}{k}=0$ is left for you to consider on your own.