Ellipses The Second of Our Conics The Second of Our Patterns (10.2)

POD At tables, write down everything you know about ellipses. Then let’s combine it here.

Start ellipses with some hands on From each table, come and get a styrofoam board, handout, piece of yarn, and two tacks. Be careful with my supplies! Follow the verbal instructions (no handout today– we’ll save a tree). In the end, what have we generated?

Based on our handiwork What is a reasonable definition for an ellipse? What are definitions of the following? Locate them on your sketch: centermajor axis (radius) (a) fociminor axis (radius) (b) verticesx- and y-radius focal radius (c)

Relationships There is a special relationship between the lengths of a, b, and c. Do you remember what it is?

Relationships There is a special relationship between the lengths of a, b, and c. Do you remember what it is? (focal radius) 2 + (minor radius) 2 = (major radius) 2 In other words,

Relationships There is another special relationship between a and c. The ratio between the length from center to focus, and the length from center to vertex is called the eccentricity (e) of the ellipse. In other words, e = c/a. Eccentricity ranges from 0 to 1. Which of these would be with a flatter ellipse, and which a rounder ellipse?

Equations Just as we have a vertex form for parabolas, with (h, k) as the vertex, we have a form for ellipses with (h, k) as the center. Remember it?

Equations Just as we have a vertex form for parabolas, with (h, k) as the vertex, we have a form for ellipses with (h, k) as the center. Remember it? What do the variables represent?

Equations There is a second, general equation for quadratic relations. For us for now, B = 0. What happens if A = 0, or C = 0? What happens if A = C? What condition exists for an ellipse?

Equations We can graph from either. We can also move from one type of equation to the other with some algebra.

Use it 1. Find the pertinent bits for this ellipse. Which way is it oriented? center x- and y-radius focal radius vertices foci

Use it 1. Find the pertinent bits for this ellipse. Which way is it oriented? center x- and y-radius focal radius vertices foci

Use it 1. Find the pertinent bits for this ellipse. Which way is it oriented? center: (0, 0) x- and y-radius: 3 and √2 focal radius: √7 vertices: (3, 0) and (-3, 0) (It’s horizontal.) foci: (√7, 0) and (- √7, 0)

Use it 1. Find the pertinent bits for this ellipse. Sketch it. center: (0, 0) x- and y-radius: 3 and √2 focal radius: √7 vertices: (3, 0) and (-3, 0) (It’s horizontal.) foci: (√7, 0) and (- √7, 0)

Use it 1. Find the pertinent bits for this ellipse. Ooh, sketch it on calculators. Do the parts match? center: (0, 0) x- and y-radius: 3 and √2 focal radius: √7 vertices: (3, 0) and (-3, 0) (It’s horizontal.) foci: (√7, 0) and (- √7, 0)

Use it 2. Find an equation for the ellipse with the center at (1, 1), a vertex at (1, -3) and a focus at (1, 3). What are the other vertex and focus?

Use it 2. Find an equation for the ellipse with the center at (1, 1), a vertex at (1, -3) and a focus at (1, 3). What are the other vertex and focus? The other focus is (1, -1) and the other vertex is (1, 5). What are the radii?

Use it 2. Find an equation for the ellipse with the center at (1, 1), a vertex at (1, -3) and a focus at (1, 3). What are the other vertex and focus? c = 2 a = 4

Use it 2. Find an equation for the ellipse with the center at (1, 1), a vertex at (1, -3) and a focus at (1, 3). What are the other vertex and focus? c = 2 a = 4 b = 2√3 Which length is horizontal and which is vertical? What is an equation for this ellipse?

Use it 2. Find an equation for the ellipse with the center at (1, 1), a vertex at (1, -3) and a focus at (1, 3). What are the other vertex and focus? c = 2 a = 4 b = 2√3

Use it 3. Graph the ellipse given by this general quadratic relationship. How do you know it’s an ellipse? (Ooh, completing the square.) Can you work this backwards?

Use it 3. Graph the ellipse given by this general quadratic relationship.

Use it 3. Graph the ellipse given by this general quadratic relationship. a = 4 (on y-axis)b = 3 center: (-2, 1)focal radius = √7

Use it 3. Graph the ellipse given by this general quadratic relationship. a = 4 (on y-axis) b = 3 center: (-2, 1) focal radius = √7

Another way to view ellipses Look at the handout. On the front are various diagrams of planes, cones and the conic intersections. On the back is a way to view conics using a focus and a directrix. Notice how eccentricity is defined here.

Another way to view ellipses