The Main Idea

Conics can also be expressed in polar form, where the focus is placed at the pole (origin). In this setting, the general form of a conic is:

\[
r = \dfrac{ed}{1 \pm e\cos\theta} \quad \text{or} \quad
r = \dfrac{ed}{1 \pm e\sin\theta}
\]

Here:

• [latex]e[/latex] = eccentricity (determines the type of conic).

• [latex]d[/latex] = directrix constant (distance related to conic definition).

• The sign/choice of cosine or sine determines orientation: right/left (cosine) or up/down (sine).

Quick Tips: How to Identify the Conic

1. Look at the Eccentricity

   • [latex]e=1[/latex] → parabola.

   • [latex]e<1[/latex] → ellipse (if [latex]e=0[/latex], it’s a circle).

   • [latex]e>1[/latex] → hyperbola.

2. Check Orientation

   • [latex]r=\dfrac{ed}{1+e\cos\theta}[/latex] → focus at origin, directrix vertical.

   • [latex]r=\dfrac{ed}{1-e\cos\theta}[/latex] → similar but mirrored.

   • [latex]r=\dfrac{ed}{1\pm e\sin\theta}[/latex] → opens upward or downward.

3. Summary Rule

   • The eccentricity [latex]e[/latex] tells you which conic.

   • The function (cos/sin) and sign tell you which direction.

The Main Idea

Conics can be graphed directly in polar coordinates using equations of the form:

\[
r = \dfrac{ed}{1 \pm e\cos\theta} \quad \text{or} \quad
r = \dfrac{ed}{1 \pm e\sin\theta}
\]

Here:

• [latex]e[/latex] = eccentricity (ellipse if [latex]e<1[/latex], parabola if [latex]e=1[/latex], hyperbola if [latex]e>1[/latex]).

• [latex]d[/latex] = directrix constant.

• The choice of cosine vs sine determines orientation (cosine = horizontal, sine = vertical).

• The sign (+/–) determines whether the conic opens right/left or up/down.

Quick Tips: How to Graph Polar Conics

1. Determine the Type of Conic

   • Compute [latex]e[/latex].

   • [latex]e<1[/latex] → ellipse, [latex]e=1[/latex] → parabola, [latex]e>1[/latex] → hyperbola.

2. Find the Orientation

   • Cosine in denominator → horizontal axis.

   • Sine in denominator → vertical axis.

   • Plus/minus sign tells whether it opens left/right or up/down.

3. Plot Key Points

   • At [latex]\theta=0, \pi/2, \pi, 3\pi/2[/latex], calculate [latex]r[/latex].

   • Plot these points carefully in polar coordinates.

4. Sketch the Curve Using Symmetry

   • Conics in polar form are often symmetric about an axis.

   • Plot enough points to capture curvature, then reflect across symmetry.

The Main Idea

All conic sections—parabolas, ellipses, and hyperbolas—can be defined in terms of a focus (a fixed point) and a directrix (a fixed line). A conic is the set of all points [latex]P[/latex] in the plane such that the ratio of the distance from [latex]P[/latex] to the focus and the distance from [latex]P[/latex] to the directrix is a constant [latex]e[/latex], called the eccentricity.

• If [latex]e=1[/latex], the conic is a parabola.

• If [latex]e<1[/latex], the conic is an ellipse.

• If [latex]e>1[/latex], the conic is a hyperbola.

This definition unifies all the conics under one geometric property.

Quick Tips: Understanding Focus-Directrix Definition

1. General Definition

   • A conic is all points [latex]P[/latex] such that [latex]e=\dfrac{\text{distance from P to focus}}{\text{distance from P to directrix}}[/latex].

2. Special Cases by Eccentricity

   • Parabola: Distance to focus = distance to directrix ([latex]e=1[/latex]).

   • Ellipse: Distance to focus < distance to directrix ([latex]e<1[/latex]).

   • Hyperbola: Distance to focus > distance to directrix ([latex]e>1[/latex]).