Conic SectionsHyperbolas

DefinitionThe conic section formed by a plane which intersects both of the right conical surfacesFormed when or when the plane is parallel to the axis of the cone

DefinitionA hyperbola is the set of all points in the plane whereThe difference between the distancesFrom two fixed points (foci)Is a constantGeogebraDemonstration

Experimenting with DefinitionTurn 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 EllipseTransverse axisLine joining the interceptsConjugate axisPasses through center, perpendicular to transverse axisVerticesPoints where hyperbola intersects transverse axis

Elements of An EllipseTransverse AxisLength = 2aFociLocation (-c, 0), (c, 0)

AsymptotesExperiment with Pythagorean relationship

Equations of An EllipseGiven equations of ellipse Centered at originOpening right and leftEquations of asymptotes

Opening up and downEquations of asymptotes

Try It OutFindThe centerVerticesFociAsymptotes

More Trials and TribulationsFind 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

AssignmentHyperbola AExercise set 6.3Exercises 1 25 odd and 33 45 odd

Conic Sections

Eccentricity of a HyperbolaThe hyperbola can be wide or narrow

Eccentricity of a HyperbolaAs with eccentricity of an hyperbola, the formula is

Note that for hyperbolas c > aThus eccentricity > 1

Try It OutIf the vertices are (1, 6) and (1, 8) and the eccentricity is 5/2Find 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 = 2Find two possible hyperbola equations

Application Locating PositionFor any point on a hyperbolic curveDifference 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 PositionThe 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 transmittersRepeat the procedureFind another curve representing the craft's position.

ConstructionConsider a the blue stringKeep marker against ruler and with string tightKeep end of ruler on focus F1 , string tied to other end

Graphing a Hyperbola on the TIAs with the ellipse, the hyperbola is not a functionPossible to solve for yGet two expressionsGraph each

What happens if it opens right and left?

Graphing a Hyperbola on the TITop and bottom of hyperbola branches are graphed separately

As with ellipses you mustEllipse with Geogebra

AssignmentHyperbolas 2Exercise Set 6.3Exercises 27 31 odd and 49 63 odd