Conic Sections The Ellipse Part A

Ellipse Another conic section formed by a plane intersecting a cone Ellipse formed when

Definition of Ellipse Set of all points in the plane … ___________ of distances from two fixed points (foci) is a positive _____________

Definition of Ellipse Definition demonstrated by using two tacks and a length of string to draw an ellipse

Note various parts of an ellipse Graph of an Ellipse Note various parts of an ellipse

Deriving the Formula Note Why? Write with dist. formula Simplify

Major Axis on y-Axis Standard form of equation becomes In both cases Length of major axis = _______ Length of __________ axis = 2b

Using the Equation Given an ellipse with equation Determine foci Determine values for a, b, and c Sketch the graph

Find the Equation Given that an ellipse … What is the equation? Has its center at (0,0) Has a minor axis of length 6 Has foci at (0,4) and (0,-4) What is the equation?

Ellipses with Center at (h,k) When major axis parallel to x-axis equation can be shown to be

Ellipses with Center at (h,k) When major axis parallel to y-axis equation can be shown to be

Find Vertices, Foci Given the following equations, find the vertices and foci of these ellipses centered at (h, k)

Find the Equation Consider an ellipse with What is the equation? Center at (0,3) Minor axis of length 4 Focci at (0,0) and (0,6) What is the equation?

Assignment Ellipses A 1 – 43 Odd

Conic Sections Ellipse The Sequel

Eccentricity A measure of the "roundness" of an ellipse not so round very round

Eccentricity Given measurements of an ellipse Eccentricity c = distance from center to focus a = ½ the length of the major axis Eccentricity

Eccentricity What limitations can we place on c in relationship to a? _________________ What limitations does this put on When e is close to 0, graph __________ When e close to 1, graph ____________

Finding the Eccentricity Given an ellipse with Center at (2,-2) Vertex at (7,-2) Focus at (4,-2) What is the eccentricity? Remember that

Using the Eccentricity Consider an ellipse with e = ¾ Foci at (9,0) and (-9,0) What is the equation of the ellipse in standard form?

Acoustic Property of Ellipse Sound waves emanating from one focus will be reflected Off the wall of the ellipse Through the opposite focus

Whispering Gallery At Chicago Museum of Science and Industry The Whispering Gallery is constructed in the form of an ellipsoid, with a parabolic dish at each focus. When a visitor stands at one dish and whispers, the line of sound emanating from this focus reflects directly to the dish/focus at the other end of the room, and to the other person!

Elliptical Orbits Planets travel in elliptical orbits around the sun Or satellites around the earth

Elliptical Orbits Perihelion Aphelion Mean Distance Distance from focus to ________________ Aphelion Distance from _______ to farthest reach Mean Distance Half the ___________ Mean Dist

Elliptical Orbits The mean distance of Mars from the Sun is 142 million miles. Perihelion = 128.5 million miles Aphelion = ?? Equation for Mars orbit? Mars

Assignment Ellipses B 45 – 63 odd

Conic Sections Ellipse Part 3

Additional Ellipse Elements Recall that the parabola had a directrix The ellipse has _________ directrices They are related to the eccentricity Distance from center to directrix =

Directrices of An Ellipse An ellipse is the locus of points such that The ratio of the distance to the nearer focus to … The distance to the nearer directrix … Equals a constant that is less than one. This constant is the _______________.

Directrices of An Ellipse Find the directrices of the ellipse defined by

Additional Ellipse Elements The latus rectum is the distance across the ellipse ______________________ There is one at each focus.

Latus Rectum Consider the length of the latus rectum Use the equation for an ellipse and solve for the y value when x = c Then double that distance

Try It Out Given the ellipse What is the length of the latus rectum? What are the lines that are the directrices?

Graphing An Ellipse On the TI Given equation of an ellipse We note that it is not a function Must be graphed in two portions Solve for y

Graphing An Ellipse On the TI Use both results

Area of an Ellipse What might be the area of an ellipse? If the area of a circle is …how might that relate to the area of the ellipse? An ellipse is just a unit circle that has been stretched by a factor A in the x-direction, and a factor B in the y-direction

Area of an Ellipse Thus we could conclude that the area of an ellipse is Try it with Check with a definite integral (use your calculator … it’s messy)

Assignment Ellipses C Exercises from handout 6.2 Also find areas of ellipse described in 73 and 79