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 such that , where is distance to the focus, is perpendicular distance to the directrix, and is eccentricity.
- A parabola has eccentricity , so every point satisfies .
- An ellipse has eccentricity , so every point is closer to the focus than the scaled directrix distance.
- A hyperbola has eccentricity , so the focus-distance ratio is greater than .
- For a parabola with vertex at and focus , the directrix is and the equation is .
- For a parabola with vertex at and focus , the directrix is and the equation is .
- For a conic with focus at the pole and vertical directrix, a common polar form is or depending on the directrix direction.
- The eccentricity of an ellipse is , and the eccentricity of a hyperbola is also , where is the focus distance from the center and 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 .
- Focus
- A fixed point used to define a conic by measuring the distance from any point on the curve.
- Directrix
- A fixed line used to define a conic by measuring the perpendicular distance from a point to the line.
- Eccentricity
- The constant ratio 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 and 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 must be the perpendicular distance from the point to the line.
- Classifying as an ellipse or hyperbola is wrong because always gives a parabola in the focus-directrix definition.
- Forgetting that can be negative is wrong because the sign of determines whether a parabola opens left, right, up, or down.
- Mixing up , , and is wrong because eccentricity uses for ellipses and hyperbolas, not .
- Choosing the wrong polar sign is wrong because and place the directrix on opposite sides of the pole.
Practice Questions
- 1 A conic has eccentricity with focus and directrix . Identify the conic and write its equation.
- 2 Classify each conic by eccentricity: , , and .
- 3 For an ellipse with and , find the eccentricity 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.
Understanding Focus-Directrix Definition of Conics Reference
The directrix is more than a line drawn beside the curve. It provides a reference distance that changes from point to point. To test whether a point belongs to a conic, measure the straight-line distance from that point to the focus.
Then measure the shortest distance to the directrix. The shortest distance always meets the directrix at a right angle. This detail matters because a slanted measurement gives the wrong result.
The eccentricity sets how those two distances must compare. It controls the curve's openness and overall shape.
A useful way to build a parabola is to start with a focus and a directrix, then mark points that are equally far from both. The vertex lies halfway between the focus and the directrix, measured along the line perpendicular to the directrix. That line is the axis of symmetry.
Points on opposite sides of the axis occur in matching pairs. In the equation y squared equals four p x, the value p is the signed distance from the vertex to the focus. A positive p means the graph opens right.
A negative p means it opens left. For x squared equals four p y, the sign of p instead determines whether the graph opens up or down.
For ellipses and hyperbolas, the focus-directrix rule explains features that can seem separate in standard form. An ellipse has two foci, even though one focus and its matching directrix can describe the same curve. Its eccentricity measures how stretched it is.
A value near zero gives a shape close to a circle. A value closer to one gives a longer, narrower ellipse. A hyperbola has two branches because its distance condition can be met on two separated regions of the plane.
Its asymptotes show the directions that the branches approach farther from the center. They are not part of the hyperbola, so a branch never crosses one.
Polar conic equations become useful when the focus is placed at the pole, which is the origin of the polar grid. The value r gives distance from the pole, while theta gives direction. In a form such as r equals e times d divided by one plus e times cosine theta, d represents the focus-to-directrix spacing.
The plus or minus choice depends on which side of the pole the directrix lies. Watch the denominator carefully. When it becomes zero, r has no finite value.
For a hyperbola, those directions lead toward asymptotes. In applications, parabolic reflectors send parallel incoming rays toward a focus, which is why their shape appears in satellite dishes and some headlights. Elliptical paths appear in orbital models, where the location of a focus has physical meaning rather than being only a graphing feature.