Circles fall into the category of ellipses. The difference between a true ellipse and a circle is the eccentricity. Eccentricity is a ratio of c/a. It ranges from 0 to 1. The closer the eccentricity to 1, the longer it will be, and more like an ellipse. However, as it nears 0, the more circular the ellipse will be.
Circles have an eccentricity of 0. This allows a and b to be the same. And is therefore, what differs them from a true ellipse.
In a circle a will always equal b. And both of these distances will be the radius.
As a result, the general form of the equation for a circle will be
This comes from a formula related to that of an ellipse, which is the first formula shown in the image. Since both squares are being divided by the same number, this equation can be simplified to
The second formula is equaled to r squared because in a circle, the distance from the center to any point will be the same, and this point is called the radius. a is the distance of the radius, so a squared will become r squared, just like in the second formula of Figure 1.
How to solve for a circle? How to graph a circle?
Click on the links to watch videos on how to solve for a circle and graph them.
Eccentricity Problems
We know eccentricity is the ratio of c to a, (c/a). So, when given the values of a, b and c, all we have to do is to replace c and a with their corresponding values.
There may be times where word problems are given. Therefore, we must know some of the vocabulary used in them.
Example – A near-sighted eye has a depth of 25 mm and an eccentricity of 0.39. What is the length of the eye?
Here the depth will always be equal to 2a and the length to 2b. We have the eccentricity, which is c/a. And we also have a, all we have to do is solve for a.
25mm = 2a
So we divide both sides by 2, and we have that a has a value of 12.5 mm. Now we substitue those values to the eccentricity formula and solve for c, as shown in Figure 4.
The Conic Club 