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Conic Sections

Conic Sections — Definitions, Locus, and General Equation

General Second-Degree Equation of a Conic

Any conic section can be represented by the general quadratic equation:

The presence of the $Bxy$ term indicates a possible rotation of axes. Circles, ellipses, parabolas, and hyperbolas can all appear rotated when $B \neq 0$.

How to Distinguish Conics Using A, B, and C

Ignoring rotation (or after rotating axes so that $B = 0$), the nature of the conic is determined by the signs of $A$ and $C$:

• Circle: $A = C$ and $B = 0$
• Ellipse: $A$ and $C$ have the same sign but not equal
• Parabola: either $A = 0$ or $C = 0$ (but not both)
• Hyperbola: $A$ and $C$ have opposite signs

With rotation included ($B \neq 0$), the classification is done using the discriminant:

• Ellipse: $\Delta < 0$ (includes circles)
• Parabola: $\Delta = 0$
• Hyperbola: $\Delta > 0$

Eccentricity of Conic Sections

The eccentricity of a conic section describes the shape and “flatness” of the curve. It is defined as:

This value fully determines the nature of the conic:

• Circle: $e = 0$
• Ellipse: $0 < e < 1$
• Parabola: $e = 1$
• Hyperbola: $e > 1$

1. Eccentricity of an Ellipse

For the ellipse $$ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1, $$ the first eccentricity is:

The eccentricity describes flatness:

• $e = 0$: ellipse becomes a circle
• small $e$: round ellipse
• $e$ close to 1: very flat, elongated ellipse

2. Second Eccentricity of an Ellipse

An alternate measure of shape is the second eccentricity:

Using $c^2 = a^2 - b^2$, we may also write:

3. Distance From Center to the Directrix of an Ellipse

The directrices of the ellipse lie at the distance:

measured horizontally from the center.

Derivation

Start with the eccentricity definition:

A point $P$ on the ellipse satisfies:

Evaluating this at the vertex $(a,0)$ gives:

which simplifies to confirm the directrix location:

Thus: