Conic Sections

Conic sections are different types of graphs that are derived from different cross sections of a double cone. There are four types of conic sections: circle, ellipse, parabola, and hyperbola.

All conic section equations are derived from the standard form equation , where A, B, C, D, and F are constants.

Terminology

Foci and Directrix

Every conic section is created about a specific fixed point or points, called the focus or foci (for plural).

The directrix is a line or lines used to construct a conic section.

The distance from any point on the conic section to the focus is proportional to the distance from that same point to the directrix.

Eccentricity

The eccentricity (e) of a conic section can be thought of as a measure of how much the shape of a conic section deviates from being perfectly circular.

Here are the eccentricities of the different conic sections:

• Circle: e = 0 (since it is perfectly circular)
• Ellipse: 0 < e < 1
• Parabola: e = 1
• Hyperbola: e > 1

We can solve for the exact eccentricities of ellipses and hyperbolas using the following equation:

c = distance from the center to a focal point

a = distance from the center to a vertex

Circles

Ellipses

Horizontal Ellipse

Vertical Ellipse

a is the distance from the center (h, k) to the vertex

b is the distance from the center to the co-vertex

c is the distance from the center to the focus

2a = major axis (the distance from vertex to vertex)

2b = minor axis (the distance from covertex to covertex)

a² is always the bigger number!

We can solve for a, b, and c using a “modified” Pythagorean Theorem:
a² = b² + c²

Parabolas

Latus rectum – a chord that passes through the foci and is perpendicular to the axis of symmetry. The length of the latus rectum is 4 times the focal length.

The vertex is at point (h, k).

P is the distance between the focus and the vertex AND the distance between the vertex and the directrix.

Hyperbolas

A box can be drawn connecting the vertices with perpendicular points on the asymptotes, as seen above.

2a =  length of transverse axis (distance between vertices)

2b = length of conjugate axis (height of the box)

c = distance from center to focal point

We can solve for a, b, and c by using the equation a² + b² = c². This is different than the equation for ellipses, since the foci are located past the vertex in hyperbolas.

General Form to Standard Form

The standard form of the conic section equations is the form that is used to graph conic sections and easily determine information. By looking at the standard form of a conic section, we can easily tell the shape of a conic section.

• The equation of a circle does not have a and b values, or a and b are the same.
• The equation of an ellipse have different a and b values, and will have an addition sign.
• The equation of a parabola will only have one squared variable (x or y).
• The equation of a hyperbola will have a subtraction sign.

The general form of a conic section equation looks like

We can change a general form equation to the more useful standard form by completing the square.

Practice Problems

1. Find the vertices and center of the ellipse .
2. Determine the coordinates of the foci of the ellipse  .

Questions 3 and 4 refer to the hyperbola below:

3. Write the equation for the hyperbola if b = 1.

4. Which of the following could be an equation of one of the asymptotes?

A. 

B. 

C. 

D. 

5. What does the graph of  look like?

A. Circle

B. Parabola

C. Ellipse

D. Hyperbola

Answers

1. Center: (5, 3). Vertices: (9, 3), (1, 3).
2. (10, 3) and (-14, 3)
3. 
4. B
5. D