An ellipse is a fascinating geometric shape defined by a simple but powerful property. It consists of all points where the sum of distances to two fixed points, called foci, remains constant. Let me show you how this works with an animation.
Notice how the sum of distances remains constant as the point moves around the ellipse. Now let me show you the key components: the major axis is the longest diameter, the minor axis is the shortest diameter, and the center is the midpoint between the foci.
Now let's transform our geometric understanding into algebraic form. The standard equation of an ellipse centered at the origin is x squared over a squared plus y squared over b squared equals one.
Here, a represents the semi-major axis length, b represents the semi-minor axis length, and c is the focal distance. These parameters are related by the equation c squared equals a squared minus b squared.
Let's verify this equation by testing different points on the ellipse. Watch how every point on the ellipse satisfies the equation, with the result always equaling one.
Now let's see how changing the parameters a and b affects the shape of our ellipse. Parameter a controls the horizontal stretch, while parameter b controls the vertical stretch.
Watch what happens when we increase parameter a. The ellipse stretches horizontally, becoming wider while maintaining the same height.
Now let's change parameter b. As b increases, the ellipse stretches vertically, becoming taller. When a equals b, we get a perfect circle.
Finally, let's see extreme cases. When a is much larger than b, we get a very flat ellipse. When b is much larger than a, we get a tall, narrow ellipse.
So far we've looked at ellipses centered at the origin. But what happens when we move the ellipse to a different location? This is called translation, and it changes our equation in a predictable way.
The original equation x squared over a squared plus y squared over b squared equals one becomes x minus h squared over a squared plus y minus k squared over b squared equals one, where h and k are the coordinates of the new center.
Let's see this in action. Watch as I translate the ellipse from the origin to the point (2, 1). Notice how the center coordinates appear directly in the equation.
Let me demonstrate with another example. I'll move the center to different positions, and you can see how the h and k values in the equation change accordingly.
Ellipses are not just mathematical curiosities - they appear everywhere in our world. The most famous example is planetary motion. Earth and all planets orbit the Sun in elliptical paths, with the Sun located at one of the two foci.
Watch as Earth moves along its elliptical orbit. The equation for this orbit can be written in standard form, and we can calculate its eccentricity, which measures how stretched the ellipse is.
Another important application is in satellite dishes. These use the reflective property of ellipses - signals coming from space reflect off the elliptical surface and converge at the focus, where the receiver is placed.
The eccentricity formula e equals c over a tells us about the shape. When e equals zero, we have a perfect circle. As e approaches one, the ellipse becomes more stretched. Understanding these relationships helps engineers design everything from telescope mirrors to architectural structures.