What is the geometric shape of the following equation:

$$
x^{2}+2 y^{2}-10 x+12 y+43=0
$$
Our equation can be written in the following form:

$$
(x-5)^{2}-25+2(y+3)^{2}-18+43=0
$$

which is

$$
(x-5)^{2}+2(y+3)^{2}=0
$$

This equation can only hold if

$$
x-5=0 \text { and } y+3=0
$$

that is, our equation is only satisfied by the coordinates of a single point: $x=5, y=-3$. The geometric image of the equation is therefore the point (5, $-3$).

István Kondor (Bencés g. VIII. o. Kőszeg)

Note. In this case, the point $(5,-3)$ can be conceived as if it were the center of an ellipse whose axes have also shrunk to this point.

If we want to maintain the assertion that the given equation is that of a conic section, or a second-order curve, then we can say that we are dealing with a degenerate conic section that has become an imaginary pair of lines.