Accelerated Precalculus Ellipses

One Minute Question Find the diameter of: x 2 + y 2 + 6x - 14y + 9 = 0

One Minute Question Find the diameter of: x 2 + y 2 + 6x - 14y + 9 = 0 x 2 + 6x y y + 49= 49 So the radius is 7 and the diameter is 14.

Questions on Homework?

Check so far... Write the equation of the parabola with focus at (4, 6) and directrix at y = -2. (1/16)(x - 4) 2 = (y - 2) or (x - 4) 2 = 16(y - 2)

Check so far... Write the focus, vertex, and directrix of: y 2 = 4y + 2x + 8 y 2 - 4y + 4 = 2x (y - 2) 2 = 2(x + 6) ½(y - 2) 2 = (x + 6)

Check so far... Write the focus, vertex, and directrix of: ½(y - 2) 2 = (x + 6) The vertex is (-6, 2). It opens to the right. So the focus is (-5.5, 2) and the directrix is x = -6.5

Check so far... Write the equation of a circle whose diameter has endpoints at (4, 5) and (-2, 9). The midpoint is (1, 7) The radius is  =  13 So... (x - 1) 2 + (y - 7) 2 = 13

The Ellipse ellipse fociAn ellipse is the set of all points P in a plane, so that the sum of the distances from P to two fixed points in the plane (called foci) is a constant.

The Ellipse Place two pushpins on your graph paper at horizontal or vertical lattice points. These will be your foci.

The Ellipse Cut a piece of string longer than the distance between your points to become your constant sum of lengths.

The Ellipse Tie each end of the string to the push pin and, using your pencil to stretch the string as far as you can, draw the ellipse.

The Ellipse The center of the ellipse is the midpoint of your foci. The segment joining two points on the ellipse that contains the foci is called the major axis. Measure your major axis.

The Ellipse The segment joining two points on the ellipse that is perpendicular to the major axis is called the minor axis. Find the length of your minor axis.

The Ellipse We call the distance from the center to a focus the focal length (c), the distance from the center to an endpoint of the major axis the semimajor axis (a) and the distance from the center to an endpoint of the minor axis the semiminor axis (b).

The Ellipse b c a

The Ellipse Pick up your string and note that its length is the length of the major axis. Then put the string back in its original place and make it taut at the endpoint of the minor axis.

The Ellipse See that b 2 + c 2 = a 2 b a c a

The Ellipse The algebra… This is basically a circle with two different radii - one in the x direction, and one in the y direction. So, if (x - h) 2 + (y - k) 2 = r 2 for a circle...

The Ellipse if you divide both sides by r. Make the two radii different and...

The Ellipse But a and b are the radii, so or

The Ellipse If your ellipse is centered at the origin, write its equation and verify that b 2 + c 2 = a 2.

The Ellipse Given: sketch it, find the center, foci, endpoints of the major axis (vertices) and endpoints of the minor axis (co-vertices.)

The Ellipse Given: Center: (3, -1), Foci: (-1,-1) and (7,-1) Vertices: (-2, -1) and (8, -1) Co-vertices: (3, 2) and (3, -4)

The Ellipse Given: 4x 2 + 9y 2 + 6x - 8y = 11 Write it in standard form, sketch it, find the center, foci, endpoints of the major axis (vertices) and endpoints of the minor axis (co-vertices.)

The Ellipse Given: 4x 2 + 9y 2 + 6x - 18y = 11 4(x 2 + (3/2)x + (9/16)) + 9(y 2 – 2y + 1) = 11 +9/ (x + ¾) 2 + 9(y – 1) 2 = 89/4 (x + ¾) 2 + (y – 1) 2 = 1 (89/16) (89/36)

The Ellipse (x + ¾) 2 + (y – 1) 2 = 1 (89/16) (89/36) The center is (-3/4, 1) The vertices are ((3 ± √89)/4, 1) The foci are ((9 ± √445)/12, 1) The co-vertices are: (3/4, (6 + √89)/6)

The Ellipse (Proof) Let the ellipse be centered at the origin with foci located at (c, 0) and (-c, 0). Let (x, y) be any point on the ellipse. By the definition of the ellipse, the sum of the distances from (x, y) to (c, 0) and (-c, 0) is the constant 2a.

The Ellipse (Proof)

The Ellipse (Proof (cont.))

But since a 2 – c 2 = b 2