If the y term was positive and the x term had a negative sign, then the hyperbola would open upwards and downwards,

Right?

A point on the solution curve

So hopefully you're now satisfied that the focal length of a hyperbola is the sum of these two denominators.

Select a point on the curve...

Start transport from this point of the curve

And it's always useful, I think, to draw that little da and then figure out OK, I'm summing to dy first.

So the curve is the upper bound when we go upwards.

This is the asymptote.

Evolute of a curve

Tangent to This Curve

Directrix of a Conic

The directrix line of a conic.

Asymptotes of a Hyperbola

So as you see here, if I charge 2.80 and I only sell 300,000, then this is my return on asset right here.

curve

Curve

Curved

Evolute of this curve

A nice visual composition of symmetry.

Center of Curvature of This Curve

Another way to think about it, in this case, when the hyperbola is a vertical hyperbola, where it opens up and down, you notice x could be equal to 0, but y could never be equal to 0.

Or you could say the difference in y over difference in x.

Let's draw some axes right there.

And also, even though it's plotted on a log curve on the left, it's curving upwards.

And actually, just so you know, this equation right here, this is a particular hyperbola that opens to the left and the right.

There you go.

You go 1 2 3 minus 3, minus 1.

Tangent to This Conic

Tangent to This Cubic Curve

but rather it had a hyperbolic orbit and was shooting through our solar system.

Splines

Spookiness

Hyperbola

Curve Mode

Bezier Curve

That's a p.

He arranged that piano music for the violin.

In particular, it could invent things like a curve which fills the plane.

And this is the case for most goods we call things like this, when income goes up, demand goes up, whole curve shifts to the right income goes down, demand goes down, whole curve shifts to the left

The two asymptotes of a hyperbola.

I have a feeling I might be running out of time.

Center of Curvature of This Conic

Center of Curvature of This Cubic Curve

Let's see if we can learn a thing or two about the hyperbola.