Focus-Directrix Definition of Conics Reference Cheat Sheet

A printable reference covering focus-directrix definitions, eccentricity, parabolas, ellipses, hyperbolas, and polar conic equations for grades 10-12.

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The focus-directrix definition of conics describes curves using distances instead of only standard equations. This cheat sheet helps students connect parabolas, ellipses, and hyperbolas under one rule involving a focus, a directrix, and eccentricity. It is useful for graphing, identifying conic types, and understanding why different conics have different shapes.

Key Facts

A conic is the set of points $P$ such that $\frac{PF}{PD}=e$ , where $PF$ is distance to the focus, $PD$ is perpendicular distance to the directrix, and $e$ is eccentricity.
A parabola has eccentricity $e=1$ , so every point satisfies $PF=PD$ .
An ellipse has eccentricity $0<e<1$ , so every point is closer to the focus than the scaled directrix distance.
A hyperbola has eccentricity $e>1$ , so the focus-distance ratio is greater than $1$ .
For a parabola with vertex at $(0,0)$ and focus $(p,0)$ , the directrix is $x=-p$ and the equation is $y^2=4px$ .
For a parabola with vertex at $(0,0)$ and focus $(0,p)$ , the directrix is $y=-p$ and the equation is $x^2=4py$ .
For a conic with focus at the pole and vertical directrix, a common polar form is $r=\frac{ed}{1+e\cos\theta}$ or $r=\frac{ed}{1-e\cos\theta}$ depending on the directrix direction.
The eccentricity of an ellipse is $e=\frac{c}{a}$ , and the eccentricity of a hyperbola is also $e=\frac{c}{a}$ , where $c$ is the focus distance from the center and $a$ is the vertex distance from the center.

Vocabulary

Conic section: A curve formed by points whose distances from a focus and a directrix have a constant ratio $e$ .
Focus: A fixed point used to define a conic by measuring the distance $PF$ from any point $P$ on the curve.
Directrix: A fixed line used to define a conic by measuring the perpendicular distance $PD$ from a point $P$ to the line.
Eccentricity: The constant ratio $e=\frac{PF}{PD}$ that determines whether a conic is a parabola, ellipse, or hyperbola.
Vertex: A turning point of a conic, often located midway between a parabola's focus and directrix.
Polar equation: An equation using $r$ and $\theta$ to describe a curve by distance from a pole and angle from a polar axis.

Common Mistakes to Avoid

Using distance to the directrix along a slanted path is wrong because $PD$ must be the perpendicular distance from the point to the line.
Classifying $e=1$ as an ellipse or hyperbola is wrong because $e=1$ always gives a parabola in the focus-directrix definition.
Forgetting that $p$ can be negative is wrong because the sign of $p$ determines whether a parabola opens left, right, up, or down.
Mixing up $a$ , $c$ , and $e$ is wrong because eccentricity uses $e=\frac{c}{a}$ for ellipses and hyperbolas, not $e=\frac{a}{c}$ .
Choosing the wrong polar sign is wrong because $r=\frac{ed}{1+e\cos\theta}$ and $r=\frac{ed}{1-e\cos\theta}$ place the directrix on opposite sides of the pole.

Practice Questions

1 A conic has eccentricity $e=1$ with focus $(3,0)$ and directrix $x=-3$ . Identify the conic and write its equation.
2 Classify each conic by eccentricity: $e=\frac{2}{3}$ , $e=1$ , and $e=\frac{5}{2}$ .
3 For an ellipse with $a=10$ and $c=6$ , find the eccentricity $e$ and explain whether the ellipse is relatively round or stretched.
4 Explain why increasing eccentricity changes a conic from ellipse to parabola to hyperbola in the focus-directrix definition.

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