16. Parametric Equations & Polar Coordinates

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

65–68. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices.

A hyperbola with vertices (0, ±2) and directrices y = ±1

Graph the ellipse $\(\frac{\left(x-1\right)^2}{9}\)+\(\frac{\left(y+3\right)^2}{4}\)=1$.

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

Given the hyperbola $\(\frac{x^2}{25}\)-\(\frac{y^2}{9}\)=1$, find the length of the $a$-axis and the $b$-axis.

53–57. Conic sections

c. Find the eccentricity of the curve.

x²/4 + y²/25 = 1

69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.

x² = -6y; (-6, -6)

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

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

A vertically oriented 3D cone is sliced with a vertical 2D plane. What is the conic section that will form?

53–56. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work.

An ellipse with vertices (0, ±9) and eccentricity ¼ 

Find the standard form of the equation for an ellipse with the following conditions.

Foci = $\(\left\)(-5,0\(\right\)),\(\left\)(5,0\(\right\))$

Vertices = $\(\left\)(-8,0\(\right\)),\(\left\)(8,0\(\right\))$

Determine if the equation $x^3+y^2+4x-8y+4=0$ is a circle, and if it is, find its center and radius.

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.

10x² - 7y² = 140

If a parabola has the focus at $\(\left\)(0,-1\(\right\))$ and a directrix line $y=1$, find the standard equation for the parabola.