Ellipses

Standard Form of an Ellipse: (x^2/a^2) + (y^2/b^2) = 1
To Sketch an Ellipse:

1. Find the major axis. The major axis is the variable with the bigger denominator. The major axis is the longer part of the ellipse.




2. Find the minor axis. The minor axis is the variable with the smaller denominator. The minor axis is the shorter part of the ellipse.




3. Now you need the vertices. The vertices are on the major axis. It is the square root of the larger denominator in point form.




4. Find the other intercepts. The other intercepts are on the minor axis. It is the square root of the smaller denominator in point form.




5. Length of the major axis. The length of the major axis is the square root of the larger denominator times two.




6. Length of the minor axis. The length of the minor axis is the square root of the smaller denominator times two.




7. Find the foci. The foci are on the major axis. To find the foci you use the formula: (focus)^2 = (larger denominator) - (smaller denominator). Then put the foci in point form.

Example: Sketch the Ellipse

(x^2/16) + (y^2/25) = 1

1. The major axis is the y-axis because its denominator is larger.




2. The minor axis is the x-axis because its denominator is smaller.




3. The square root of 25 is +/- 5 so the vertices are (0,5) and (0,-5).




4. The square root of 16 is +/- 4 so the other intercepts are (4,0) and(-4,0)




5. The square root of 25 is 5 times 2 is 10 so the length of the major axis is 10 units.




6. The square root of 16 is 4 times 2 is 8 so the lenght of the minor axis is 8 units.




7. (focus)^2 = 25 - 16

(focus)^2 = 9

focus = +/- 3

so the foci are (0,3) and (0,-3)