๐ Definition of a Circle
A circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center. This fixed distance is called the radius.
- ๐ The distance from the center to any point on the circle is constant.
- ๐ All points on the circle lie in the same plane.
- ๐ A circle has infinite lines of symmetry.
๐ Definition of an Ellipse
An ellipse is defined as the set of all points in a plane such that the sum of the distances from two fixed points, called foci (plural of focus), is constant.
- ๐ฏ The sum of the distances from any point on the ellipse to the two foci is constant.
- ๐ All points on the ellipse lie in the same plane.
- โ๏ธ An ellipse has two axes of symmetry: the major axis and the minor axis.
๐ Circle vs. Ellipse: A Side-by-Side Comparison
| Feature |
Circle |
Ellipse |
| Definition |
Set of points equidistant from a center. |
Set of points where the sum of distances to two foci is constant. |
| Equation |
$(x-h)^2 + (y-k)^2 = r^2$ |
$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ |
| Foci |
One (the center) |
Two |
| Axes |
All diameters are equal. |
Major and minor axes of different lengths. |
| Eccentricity |
0 |
Between 0 and 1 ($0 < e < 1$) |
| Shape |
Perfectly round |
Oval |
๐ก Key Takeaways
- ๐ A circle is a special case of an ellipse where the two foci coincide at the center.
- ๐ง The equation of an ellipse generalizes the equation of a circle. When $a = b$ in the ellipse equation, it becomes the equation of a circle.
- โ๏ธ Understanding the foci and eccentricity helps to visualize and differentiate between circles and ellipses.
- โ Circles have constant radius; ellipses have a constant sum of distances to the foci.