It’s Professor Dave, let’s graph some ellipses.
We just learned how to graph circles, so let’s move on the next type of conic section, the ellipse.
Let’s recall that while a circle is the conic section formed when a plane intersects
this cone parallel to the base, an ellipse is what we get when this plane is not parallel
to the base, but it still just intersects one half of the cone.
So that’s one way to define an ellipse.
Another way is to say that an ellipse is the set of all points surrounding two foci, whereby
the distance from a point to one focus plus the distance from that point to the other
focus is constant.
No matter what point we are looking at on this ellipse, the sum of these two distances
will be the same.
As it gets closer to one, it’s farther from the other, and vice versa.
The midpoint of the line segment that connects the foci is the center of the ellipse.
Ellipses also have two axes of symmetry, a major axis, which will be the longer one,
and a minor axis, which is the shorter one.
At each end of the major axis we can find a vertex.
And that’s all the terminology we need to know.
The standard form of the equation of an ellipse that has its center at the origin looks like this.
It’s similar to the equation for a circle, but we have these terms A and B, which give
us information about the major and minor axes.
Specifically, the major axis has a length of two A, and the minor axis has a length
of two B, or we could think of A as the distance from the center to a vertex on the major axis,
which means that positive and negative A are the X intercepts and positive and negative
B are the Y intercepts.
Beyond this, we can use this equation, C squared equals A squared minus B squared, and if we
solve for C, that is the distance from the center to either of the foci.
If an ellipse has its equation in standard form, with a one over here, it’s very easy
to graph it.
Say we have a nine for A squared and a four for B squared.
That means A is three and B is two, so we would just draw the four intercepts, connect
them, and there’s the ellipse.
The foci are not part of the ellipse, but if we need those as well, that will come from
C squared equals nine minus four, or C equals plus or minus root five.
We should note that A is always greater than B, so if the term below X squared is greater
than the term below Y squared, we have A here and B here, which produces an ellipse that
is wider than it is tall.
But if the other number is larger, then this term is A squared, and this other term is
B squared.
This is because the ellipse will now be taller than it is wide, but the A term must always
correspond to the major axis, which is the longer axis, so that will now be the vertical
axis given by Y.
Let’s say we have an equation for an ellipse where this number doesn’t equal one.
We will have to divide the entire equation by this term so that we get a one on this
side, as it will be easier to graph in standard form, even if we end up with weird numbers
for A and B.
We could also graph an ellipse if given information about its foci and vertices.
Say the foci are at plus and minus one, while the vertices are at plus and minus two.
That means that C is one and A is two.
It will then be true that one equals four minus B squared, or B squared equals three.
Then we just put A squared and B squared into the standard form, and there’s our equation.
Ellipses can be transformed by some of the ways we already know, except in this case,
we can write X minus H quantity squared and Y minus K quantity squared rather than X squared
and Y squared, and these H and K values will be the horizontal and vertical shifts respectively.
Notice the minus signs, so X minus two will mean a shift of two to the right, not left.
When we have H and K terms, the ellipse has a center at the point H, K, and everything
else we learned about the axes and foci will stay the same.
That’s really all we need to know about ellipses right now.
These are extremely important shapes for astronomy and astrophysics, so if you move on to my
astronomy course, you will learn all about how celestial objects, like planets and comets,
follow elliptical orbits, rather than circular ones as originally thought many centuries ago.
But for right now, we’ve got a lot more math to cover, so let's check comprehension.