Conic Sections Review

A review of Unit 7, by Nick Hayes

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1. What Are Conic Sections?

All conic sections are derived from a double cone, the form shown below:

Conic sections are created by slicing the double cone form.

Definition of a Conic Section:

Any shape that is formed by slicing a double cone.

Knowledge of the double cone and what a conic section is is the key to learning this unit!

We'll be studying the following four types of conic sections. Be sure to pay attention!

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2. Parabolas

Q: What is a parabola?

A: A parabola is a shape formed by slicing a double cone on a slant, as shown in the example below:

A parabola’s location on the comic section is defined by the location of a point called the focus and a line called the directrix.

All of this information gives us the official definition of a parabola.

Parabola:

A parabola is the set of all points in a plane that are the same distance from a given point called the focus and a given line called the directrix.

The equation for a parabola can take two forms:

y = a(x - h)2 + k

x = a(y - k)2 + h

(h , k) represents the vertex of the parabola.

In the equations, the axis of symmetry for each is as follows:

x = h

y = k

The focus for each is as follows:

(h , k + 1/4a)

(h + 1/4a , k)

The directrix for each is as follows:

y = k - 1/4a

x = h - 1/4a

The direction of opening for each is as follows:

Upward if a > 0 ; Downward if a < 0

Right if a > 0 ; Left if a < 0

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3. Circles

Circle: 

The set of all points in a plane that are equidistant from a given point in the plane called the center.

Circles also have radii:

Any segment whose endpoints are the center and a point on the circle is a radius of the circle.

The equation of a circle is as follows:

(x - h)2 + (y - k)2 = r2

Where (h , k) is the center of the circle and r is the radius.

Circles can be formed by slicing a double cone, thus making them a conic section:

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4. Ellipses

An ellipse is another example of a conic section. Ellipses are similar to circles, except instead of a center they have two foci and can be wide or tall.

Ellipse:

The set of all points in a plane such that the sum of the distances from the foci is constant.

Equations for an ellipse:

Equation 1 represents an ellipse with a horizontal major, and equation 2 represents an ellipse with a vertical major.

Horizontal Major:

  Vertical Major:

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5. Hyperbolas

A hyberbola is our final example of a conic section. A hyperbola is similar to a parabola, but has two arches opening in opposite directions as opposed to one arch.

Hyperbola:

The pair of two symmetrical curves formed by the intersection of a plane with two equal cones on opposites of the same vertex.

The equation for a hyperbola is as follows: