Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of...

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Conic Sections Hyperbolas

Transcript of Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of...

Page 1: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

Conic Sections

Hyperbolas

Page 2: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

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

Page 3: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

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

Page 4: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

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.

Page 5: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

Elements of An Ellipse

• ____________ axis Line joining the

intercepts

• Conjugate axis Passes through

________, perpendicularto transverse axis

• Vertices Points where hyperbola _______________

transverse axis

Page 6: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

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

Page 7: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

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

Page 8: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

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

Page 9: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

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

Page 10: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

Assignment

• Hyperbola A

• Exercise set 6.3

• Exercises 1 – 25 oddand 33 – 45 odd

Page 11: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

Conic Sections

Page 12: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

Eccentricity of a Hyperbola

• The hyperbola can be wide or narrow

Page 13: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

Eccentricity of a Hyperbola

• As with eccentricity of an ellipse, the formula is

Note that for hyperbolas _________

Thus eccentricity > 1

eccentricity

Page 14: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

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

Page 15: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

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.”

Page 16: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

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.

Page 17: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

Construction

• Consider a the blue string

• Keep markeragainst rulerand with stringtight

• Keep end ofruler on focusF1 , string tied to other end

Page 18: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

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?

Page 19: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

Graphing a Hyperbola on the TI

• Top and bottom of hyperbola branches are graphed separately

As with ellipsesyou must

__________

Page 20: Conic Sections Hyperbolas. Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when _________ or.

Assignment

• Hyperbolas 2

• Exercise Set 6.3

• Exercises 27 – 31 oddand 49 – 63 odd