Lesson 25 — Conics: ellipse, parabola, hyperbola

Canonical equations

Ellipse

Sum of distances to 2 foci is constant: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Where $a > b > 0$. Foci at $(\pm c, 0)$ with $c = \sqrt{a^2 - b^2}$. Major axis $= 2a$, minor $= 2b$. Eccentricity $e = c/a \in [0, 1)$. When $a = b$, it is a circle.

Parabola

Distance to focus $=$ distance to directrix: $y^2 = 4px$

Focus at $(p, 0)$, directrix $x = -p$.

Hyperbola

Difference of distances to 2 foci is constant: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

Foci at $(\pm c, 0)$ with $c = \sqrt{a^2 + b^2}$. Eccentricity $e = c/a > 1$. Asymptotes $y = \pm (b/a) x$.

Table of the three conics

General form of a conic

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

Discriminant $\Delta = B^2 - 4AC$:

• $\Delta < 0$: ellipse (or circle if $A=C, B=0$).
• $\Delta = 0$: parabola.
• $\Delta > 0$: hyperbola.