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__________ 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

2 1PF PF k

______k

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.

Elements of An Ellipse

• ____________ axis Line joining the

intercepts

• Conjugate axis Passes through

________, perpendicularto transverse axis

• Vertices Points where hyperbola _______________

transverse axis

Elements of An Ellipse

• Transverse Axis Length = 2a

• Foci Location (-c, 0), (c, 0)

• Asymptotes Experiment with Pythagorean relationship

http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=140&CFID=388733&CFTOKEN=53107845

Equations of An Ellipse

• Given equations of ellipse Centered at origin Opening right and left Equations of

asymptotes

• Opening up and down Equations of

asymptotes

2 2

2 21

x y

a b

Try It Out

• Find The center Vertices Foci Asymptotes

2 2

125 36

x y

2 22 1

136 49

y x

2 24 25 16 50 109 0x y x y

More Trials and Tribulations

• 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

Eccentricity of a Hyperbola

• The hyperbola can be wide or narrow

Eccentricity of a Hyperbola

• As with eccentricity of an ellipse, the formula is

Note that for hyperbolas _________

Thus eccentricity > 1

eccentricity

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 ellipse

• The center of the ellipse is at (-3, -3) and the conjugate axis has length 6, and the eccentricity = 2 Find two possible ellipse 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.”

“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 ___________ 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

As with ellipsesyou must

__________

Assignment

• Hyperbolas 2

• Exercise Set 6.3

• Exercises 27 – 31 oddand 49 – 63 odd