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?

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