Locus is any system of points, lines, or curves which satisfies one or more given conditions (https://en.wikipedia.org/wiki/Locus_(mathematics)#Examples_in_plane_geometry).
We know that the locus of an equation of the second degree in two variables is a conic (i.e. an ellipse, hyperbola, or parabola).
My question is:
What is the locus of an equation of the fourth degree in two variables of the form:
$$ax^{2}y^{2}+bx^{2}y+cx^{2}+dxy^{2}+wxy+ux+vy^{2}+sy=0$$
where $a,b,c,d,w,u,v,s$ are real numbers.
This is not a general quartic curve. Its equation has the form $$Q_0(x)y^2+Q_1(x)y+Q_2(x)=0$$ where the $Q_i$ are quadratic polynomials. This can be "solved" as a quadratic in $y$ to give $$y=-\frac{-Q_1(x)+\sqrt{R(x)}}{2Q_0(x)}$$ where $R(x)$ is a polynomial of degree $\le4$. This means that the curve is birationally equivalent to the curve $y'^2=R(x)$. In general this is a genus $1$ curve, but there are degenerate cases. In your original curve $(0,0)$ is a point thereon, so in general your curve is an elliptic curve (unless it degenerates somehow).
The general quartic plane curve has genus $3$, so your curve is much more special than that.