Conic Sections. Hyperbolas. Definition. The conic section formed by a plane which intersects both of the right conical surfaces Formed when or when the plane is parallel to the axis of the cone . Definition. A hyperbola is the set of all points in the plane where

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The conic section formed by a plane which intersects both of the right conical surfaces

Formed whenor when the plane isparallel to the axis of the cone

Definition

A hyperbola is the set of all points in the plane where

The difference between the distances

From two fixed points (foci)

Is a constant

Geogebra

Demonstration

Experimenting with Definition

Turn on Explore geometric definition. A purple point will appear on the hyperbola, along with two line segments labeled L1 and L2. Drag the purple point around the hyperbola.

Do the lengths of L1 and L2 change?

What do you notice about the absolute value of the differences of the lengths?

How do these observations relate to the geometric definition of a hyperbola.

Observe the values of L1, L2, and the difference | L1 − L2 | as you vary the values of a and b.

How is the difference | L1 − L2 | related to the values of a and/or b? (Hint: Think about multiples.)

Determine the difference | L1 − L2 | for a hyperbola where a = 3 and b = 4. Use the Gizmo to check your answer.

Find the equation in standard form of the hyperbola that satisfies the stated conditions.

Vertices at (0, 2) and (0, -2), foci (0, 3) and (0, -3)

Foci (1, -2) and (7, -2)slope of an asymptote = 5/4

Assignment

Hyperbola A

Exercise set 6.3

Exercises 1 – 25 oddand 33 – 45 odd

Conic Sections

The Return of the Hyperbola

Eccentricity of a Hyperbola

The hyperbola can be wide or narrow

Eccentricity of a Hyperbola

As with eccentricity of an hyperbola, the formula is

Note that for hyperbolas c > a

Thus eccentricity > 1

Try It Out

If the vertices are (1, 6) and (1, 8) and the eccentricity is 5/2

Find the equation (standard form) of the hyperbola

The center of the hyperbola is at (-3, -3) and the conjugate axis has length 6, and the eccentricity = 2

Find two possible hyperbola equations

Application – Locating Position

For any point on a hyperbolic curve

Difference between distances to foci is constant.

Result: hyperbolas can be used to locate enemy guns

“If the sound of an enemy gun is heard at two listening posts and the difference in time is calculated, then the gun is known to be located on a particular hyperbola. A third listening post will determine a second hyperbola, and then the gun emplacement can be spotted as the intersection of the two hyperbolas.”

Application – Locating Position

The loran system navigator

equipped with a map that gives curves, called loran lines of position.

Navigators find the time interval between these curves,

Narrow down the area that their craft's position is in.

Then switch to a different pair of loran transmitters

Repeat the procedure

Find another curve representing the craft's position.

Construction

Consider a the blue string

Keep markeragainst rulerand with stringtight

Keep end ofruler on focusF1 , string tied to other end

Graphing a Hyperbola on the TI

As with the ellipse, the hyperbola is not a function

Possible to solve for y

Get two expressions

Graph each

What happensif it opensright and left?

Graphing a Hyperbola on the TI

Top and bottom of hyperbola branches are graphed separately