Can I foresee the graph of an expression only looking at the formula?

Question

If I look at the ellipse's expression $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ it will look very similar to the circle's expression $x^2 +y^2 = 1$. The only change I notice is that $a$ and $b$ are different, and if I have a bigger value of $a$ the ellipse will be flat on the x axis, different to the circle in which it has the same x, and y size on the axis!

Answer 1

Ellipse is a particular case of a conic section. Here is the general equation of conic sections: $$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. \tag{1} $$

Here is how we can distinguish the curve types by looking at the formula $(1)$:

(A) If $B^2-4AC<0$, then the curve is an ellipse or a circle or a point.

(B) If $B^2-4AC=0$, then the curve is a parabola or non-intersecting lines or one line.

(C) If $B^2-4AC>0$, then the curve is a hyperbola or two intersecting lines.