Conic Sections

Definition

Conic sections are the curves generated by the intersection of a plane with a double right circular cone. They are classified into four main types: circles, ellipses, parabolas, and hyperbolas, depending on the angle of the plane relative to the cone’s axis.

Why It Matters

These shapes describe the fundamental paths of everything from thrown baseballs to interstellar probes, making them the ‘alphabet’ of celestial mechanics.

Core Concepts

• Parabolas: Set of points equidistant from a focus and a directrix ($y^2 = 4px$).
  • How to read: “The value y squared equals four p x.”
  • Meaning: Standard right-opening parabola with vertex at origin; focus at $(p,0)$, directrix $x = -p$.
• Ellipses: Set of points where the sum of distances to two foci is constant ($\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$).
  • How to read: “The value x squared all over a squared plus y squared all over b squared equals one.”
  • Meaning: Closed oval; $a$ and $b$ are semi-axes; sum of focal distances is $2a$ when $a > b$.
• Hyperbolas: Set of points where the difference of distances to two foci is constant ($\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$).
  • How to read: “The value x squared all over a squared minus y squared all over b squared equals one.”
  • Meaning: Two open branches; difference of focal distances is constant ($2a$).
• Eccentricity ($e$): A measure of how much the conic “flattens” or “stretches.”
  • $e=0$: Circle.
  • $0 < e < 1$: Ellipse.
  • $e=1$: Parabola.
  • $e>1$: Hyperbola.
  • How to read: “The value e equals zero, e is between zero and one, e equals one, or e is greater than one.”
  • Meaning: Eccentricity classifies conics: $e=0$ circle, $0<e<1$ ellipse, $e=1$ parabola, $e>1$ hyperbola—ratio of focal distance to directrix distance.

Connected Concepts

• Conic Sections in Polar Coordinates
• Volumes Using Cross-Sections