Exercises: Derive Equations of Ellipses and Hyperbolas

What is the distance between the points $(-3, 1)$ and $(5, 7)$?

A circle is the set of all points in a plane at a fixed distance from one fixed point. Which description follows the same pattern but uses two fixed points and a constant sum?

A right triangle has legs of length $b$ and $c$ and hypotenuse of length $a$. If $a = 13$ and $c = 5$, what is $b$?

For the ellipse $\frac{x^2}{169} + \frac{y^2}{144} = 1$, find the coordinates of the two vertices (the endpoints of the major axis).

Where are the foci of the ellipse $\frac{x^2}{100} + \frac{y^2}{64} = 1$?

For the hyperbola $\frac{x^2}{144} - \frac{y^2}{25} = 1$, find the coordinates of the two foci.

What are the equations of the asymptotes of the hyperbola $\frac{x^2}{64} - \frac{y^2}{36} = 1$?

Compute the eccentricity of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$. Express your answer as a fraction in simplest form.