Conic Sections Ellipse Part 3 Additional Ellipse Elements

Conic Sections Ellipse Part 3

Additional Ellipse Elements • Recall that the parabola had a directrix • The ellipse has two 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 eccentricity.

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

Additional Ellipse Elements • The latus rectum is the distance across the ellipse at the focal point. § There is one at each focus. § They are shown in red

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 Length =

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 • Use this trick

Graphing An Ellipse On the TI • Set Zoom Square • Note gaps due to resolution • Graphing routine § Specify an x § Solve for zero of expression for y § Graph the (x, y)

Graphing Ellipse in Geogebra • Enter ellipse as quadratic in x and y

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 Exercises 69 – 74, 77 – 79 Also find areas of ellipse described in 73 and 79