91Ó°ÊÓ

Two hyperbolas that have the same set of asymptotes are called conjugate. Show that the hyperbolas \(\frac{x^{2}}{4}-y^{2}=1 \quad\) and \(\quad y^{2}-\frac{x^{2}}{4}=1\) are conjugate. Graph each hyperbola on the same set of coordinate axes.

The eccentricity \(e\) of an ellipse is defined as the number \(\frac{\mathrm{c}}{a},\) where \(a\) is the distance of a vertex from the center and \(c\) is the distance of a focus from the center. Because \(a>c,\) it follows that \(e<1 .\) Write a brief paragraph about the general shape of each of the following ellipses. Be sure to justify your conclusions. (a) Eccentricity close to 0 (b) Eccentricity \(=0.5\) (c) Eccentricity close to 1

Find the midpoint of the line segment connecting the points (3,-8) and (-2,5)