1. General Equation of an Ellipse
2. You should be familiar with the General Equation of a Circle and how to shift and stretch
graphs, both vertically and horizontally.
In this lesson, we will find the equation of an ellipse, given the graph.
3. Recall that the equation of a circle centered at the origin is x2+y2=r2
4. We can divide both sides by r2to get the right hand side equal to 1
5. and if we wish, rewrite the squares outside the parentheses. We do this to compare transfor-
mations of the circle to the transformations of waves.
6. Recall that a multiplier inside the parentheses, with the x, makes the wave thinner.
7. If instead, we divide, the wave will become wider.
8. The same is true for the circle. We can think of a circle with radius ras a circle of radius 1
that has been stretched by a factor of rboth horizontally and vertically.
9. An ellipse is a circle that has been stretched unequally. We can stretch by a factor of ain the
x-direction and a factor of bin the y-direction.
10. We can also write the equation of an ellipse without the parentheses. The last equation is the
standard form of an ellipse, centered at the origin.
11. Here is an example. This ellipse is a standard circle, centered at the origin, that has been
stretched 3 times wider and 5 times taller. You may wish to pause the video to write out the
equation.
12. The horizontal stretch is by a factor of 3, so the number under the x2is 32, which is 9, and
the number under the y2is 52, since the ellipse is stretched by a factor of 5 vertically.
13. The points on the ellipse that are farthest apart are called the major vertices and the line
segment connecting them is called the major axis. The points on the ellipse that are closest
together are called the minor vertices and the line segment connecting them is called the
minor axis. The intersection of the major axis and minor axis is called the center.
14. Now lets graph the ellipse
x2
16 +y2
36 = 1
15. This equation is in standard form. The right side of the equation is 1. 16 = 42, so the circle
has been stretched a factor of 4 in the x-direction, putting the vertices at (4,0) and (โ4,0).
16. 36 = 62so the circle has been stretched a factor of 6 in the y-direction, putting vertices at
(0,6) and (0,โ6).
17. We can now sketch the ellipse.
18. .
