16. Parametric Equations & Polar Coordinates

Determine the vertices and foci of the hyperbola $\(\frac{y^2}{4}\)-x^2=1$.

Find the equation for a hyperbola with a center at $\(\left\)(0,0\(\right\))$, focus at $\(\left\)(0,-6\(\right\))$ and vertex at $\(\left\)(0,4\(\right\))$ .

Find the equations for the asymptotes of the hyperbola $\(\frac{x^2}{64}\)-\(\frac{y^2}{100}\)=1$.

Find the equations for the asymptotes of the hyperbola $\(\frac{y^2}{16}\)-\(\frac{x^2}{9}\)=1$.

Describe the hyperbola $\(\frac{\left(x+2\right)^2}{9}\)-\(\frac{\left(y-4\right)^2}{16}\)=1$.

Describe the hyperbola $y^2-\(\frac{\left(x-1\right)^2}{4}\)=1$.

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

A parabola that opens to the right with directrix x = -4

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

A parabola with focus at (3, 0)

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

A parabola symmetric about the y-axis that passes through the point (2, -6)

How does the eccentricity determine the type of conic section?

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.

x² = 12y

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

A hyperbola with vertices (±2, 0) and asymptotes y = ±3x/2

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

A hyperbola with vertices (±4, 0) and foci (±6, 0)