1. S TANDARD F ORM OF THE E QUATION OF AN E LLIPSE

2. R EVIEW : E LLIPSES An ellipse is essentially an oval. An ellipse is a form of conic section – a curve that is defined by the intersection of a cone and a plane. An ellipse is all the points that have the same total distance from two points (foci). The longest line segment through an ellipse is its major axis. The shortest line segment is its minor axis.

3. E QUATION OF AN E LLIPSE IN S TANDARD F ORM The standard form of the equation for an ellipse is The center of this ellipse is (h, k). The two axes (major and minor) are parallel to the x and y axes. The axis parallel to the x-axis has length 2a and the axis parallel to the y-axis has length 2b. If a > b, the major axis is parallel to the x-axis. If b > a, the major axis is parallel to the y-axis.

4. I NFORMATION FROM THE S TANDARD E QUATION FOR AN E LLIPSE The standard form of the equation for an ellipse gives you quite a bit of information: Length of the major and minor axes Center of the ellipse Using the formula f 2 = a 2 – b 2 (assuming a > b), the focal length (distance from the center to the foci).

5. E XAMPLE Let’s try writing the equation of an ellipse based on some of its characteristics. What is the equation of an ellipse centered at (2, 3) whose major axis is along the x-axis and has length 4 and whose minor axis has length 2?

6. A NSWER If the major axis has length 4, a = 2. If the minor axis has length 2, b = 1. If the center is at (2, 3), h = 2 and k = 3. Thus, our equation is

7. C ONVERTING B ETWEEN THE G RAPH AND THE E QUATION It’s possible to use the information in a graph of an ellipse to find its equation. Consider the graph to the right. The ellipse is centered at (1, 3). The major axis is along the x axis and has length 8. Thus, a = 4. The minor axis is along the y axis and has length 6. Thus, b = 3. We now have everything we need to write the equation of this ellipse:

8. D ERIVATION OF THE S TANDARD E QUATION FOR AN E LLIPSE C ENTERED AT THE O RIGIN Where does the standard equation come from? We can derive it from the geometric definition of an ellipse. An ellipse is the set of points whose distances to two fixed points (called the foci) add to the same constant. Let’s define some points: (x, y) is a point on the ellipse. f 1 and f 2 are the foci (-c, 0) and (c, 0). a is half the length of the major axis. b is half the length of the minor axis. d 1 is the distance between f 1 and (x, y). d 2 is the distance between f 2 and (x, y).

9. P ICTURE

10. D ERIVATION The first thing to notice is that d 1 +d 2 = 2a. You can see this if you look at the case where (x, y) is on the x-axis. d 1 = a + c and d 2 = a – c, so d 1 + d 2 = 2a.

11. D ERIVATION

15. Before we proceed, there’s another relationship we need to prove. Consider what happens when (x, y) is on the y-axis. The distance to each of the foci is equal, in that case, so d 1 = d 2. Remember also that d 1 + d 2 = 2a, so in the case where (x,y) is on the y-axis, d 1 = d 2 = a. Notice also that the two axes form a right triangle with d 1 and d 2. Using the Pythagorean theorem, we can use these right triangles to demonstrate the relationship a 2 = b 2 + c 2 where a is half the length of the major axis, b is half the length of the minor axis, and c is the distance from the center to the foci. This relationship is shown visually on the next slide.

16. A 2 = B 2 + C 2

17. D ERIVATION

18. T RANSLATIONS Now that we’ve derived the standard equation for an ellipse centered at the origin, we can figure out logically what the equation for an ellipse centered at (h, k) would be. We know that to translate a curve horizontally, we replace x with (x – h) and if we want to translate a curve vertically, we replace y with (y – k). To move the center, we just translate the ellipse in this way. This gives us the standard equation for a general ellipse: