Question

Suppose you have an equation in two variables. How can you quickly identify it as an equation for an ellipse, hyperbola, or parabola? How might you explain that to a friend who has little math experience?

Answer 1

The equation of an ellipse is: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = c$ where: a, b - major and minor axes.

The equation of the hyperbola is: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = c$ where: a, b - major and minor axes.

The equation of the parabola is: $\mathbf{y} = \mathbf{a}\mathbf{x}^2 + \mathbf{b}\mathbf{x} + \mathbf{c}$

So, if we have any equation: $\pm a y^{n} \pm b x^{n} \pm \dots = c$ , that to identify it as parabola: one variable should be as $a y$ (the first degree only), other as $a x^{2}$ (the second degree and also can contain first degree).

If two variables is as: $a x^{2}$ (the second degree), that to identify it as ellipse or hyperbola it is necessary to consider signs of variables factors: if it's equal (+ and + or- and -) - it is ellipse, if not - hyperbola.