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March 27, 2021

Tab test

Congratulations! You've made it through the first half of the course. Pat yourself on the back for making it this far.

As of our last session, we're now done with our in-depth study of the circular functions. For the rest of the semester, we'll be jumping around between number of smaller but interconnected topics — conic sections, parametric equations, polar coordinates, vectors, and complex numbers.

That doesn't mean the circular functions are going away though.

On the contrary, we'll see them pop up every so often as just the right tool to get a better understanding of how the above topics work.

By the way, I'm trying something new with how to organize the parts of this lesson. To see the different parts of the lesson, click the buttons to open each tab.

Conic sections

(By the way, if the next set of notes seems really long, it's because there are a significant number of pictures, as we're dealing with a very visual topic. Don't panic!)

Our next topic is actually one of the most ancient objects of study in the history of geometry. When we take an infinite double cone as shown in the figure below and slice through it with a plane, the shape we get is called a conic section.

Two examples of conic sections.
Picture source: Brilliant.org

Depending on the angle of the plane compared to the side of the cone, we can get a number of different curved shapes: a circle, an ellipse, a parabola, or a hyperbola.

These shapes have been studied intensely since ancient Greece (most notably by Apollonius of Perga), no doubt in part because of their wide variety of applications such as acoustic design, non-invasive surgery, and planetary motion. They also lend themselves well to the tools of analytic geometry, as they have relatively simple algebraic equations (needing nothing more complicated than quadratics!) so we can use algebra to investigate their properties.

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Arguably the simplest type of conic section is a circle, which we get by slicing the cone at a perfectly horizontal angle.

Even when studying a simple shape like a circle, though, there's always more interesting things we can find (as you've no doubt seen this semester).

In fact, have you ever actually tried to define a circle?

It's trickier than it sounds! You might start off with talking about it being round, but there are lots of round shapes. Or you might say it has the same width all the way around, but even this isn't enough to characterize a circle. After some thought, though, you might come up with a definition like this:

A circle is a plane figure, consisting of all points that are some fixed distance (called the radius) from a given center point.

This is good so far, but we can do even better. Let's give names to some of these objects — we'll say that the radius is \(r\), the center point is \((x_0,y_0)\), and that any arbitrary point on the circle will be \((x,y)\). (Notice that this means \(x_0\), \(y_0\), and \(r\) stay the same for a given circle, while \(x\) and \(y\) can be used to describe any point on the circle.) Let's add this to our definition:

A circle is a plane figure, consisting of all points \((x,y)\) that are some fixed distance \(r\) (called the radius) from a given center point \((x_0,y_0)\).

What's nice about this is that now we can also describe it algebraically! The defining property of a circle can be written using the distance formula as follows: \[\sqrt{(x-x_0)^2+(y-y_0)^2}=r\] To avoid using ugly square roots, we can then square both sides: \[\boxed{(x-x_0)^2+(y-y_0)^2=r^2}\] This is what we'll use from now on as the equation of a circle.

Sometimes you'll also see \((h,k)\) used for the center of a circle; I personally prefer \((x_0,y_0)\) because it reminds me of the point-slope form of a line:\[y-y_0=m(x-x_0)\]Also, later on when you learn about spheres, it'll become easier to make the jump from 2D to 3D.


If a circle has its center at \((3,-2)\) and a radius of \(5\), its equation is: \[(x-3)^2+(y+2)^2=25\] In fact, if you go type this into Desmos right now, this is exactly the graph you'll see.

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If we cut our double cone at a slightly tilted angle, we'll end up with an oval shape called an ellipse.

We'll look at the algebraic equation for an ellipse in class; for now, we're just going to look at "anatomy" of an ellipse.

Unlike on a circle, the points on an ellipse aren't always the same distance from the center. The largest distance across the ellipse (the "big diameter") is called the major axis, while the smallest distance across the ellipse (the "small diameter") is called the minor axis. The respective "radii" are often called the semimajor axis and semiminor axis.

In addition, there are two special points inside an ellipse called the foci (marked below with X's).

The special property of the foci is that, given any point on the ellipse, the sum of the distances from that point to either focus always adds up to the same number, regardless of which point on the ellipse you choose.

Drag the black point around the ellipse. Try a few different points and watch what happens when you add the two distances shown.

One interesting thing about the foci of an ellipse is that if you shoot a ray from one focus of an ellipse and reflect off the rim of the ellipse, it will always bounce back to the other focus. We'll see why this is useful in class.

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