12.5 Conic Sections in Polar Coordinates

For each of the following equations, identify the conic with focus at the origin, the directrix, and the eccentricity.

1. $r = \frac{6}{3 + 2 \sin \theta}$
2. $r = \frac{12}{4 + 5 \cos \theta}$
3. $r = \frac{7}{2 - 2 \sin \theta}$

Identify the conic with focus at the origin, the directrix, and the eccentricity for $r = \frac{2}{3 - \cos \theta} .$

Find the polar form of the conic given a focus at the origin, $e = 3$ and directrix $y = - 2.$

Find the polar form of a conic given a focus at the origin, $e = \frac{3}{5} ,$ and directrix $x = 4.$

Find the polar form of the conic given a focus at the origin, $e = 1 ,$ and directrix $x = −1.$

Convert the conic $r = \frac{1}{5 - 5 \sin \theta}$ to rectangular form.

Convert the conic $r = \frac{2}{1 + 2 \cos \theta}$ to rectangular form.

Explain how eccentricity determines which conic section is given.

If a conic section is written as a polar equation, and the denominator involves $\sin \theta ,$ what conclusion can be drawn about the directrix?

What do we know about the focus/foci of a conic section if it is written as a polar equation?

Identify the conic with a focus at the origin, and then give the directrix and eccentricity.

$r = \frac{3}{4 - 4 \sin \theta}$

Identify the conic with a focus at the origin, and then give the directrix and eccentricity.

$r = \frac{5}{1 + 2 \sin \theta}$

Identify the conic with a focus at the origin, and then give the directrix and eccentricity.

$r = \frac{3}{10 + 10 \cos \theta}$

Identify the conic with a focus at the origin, and then give the directrix and eccentricity.

$r = \frac{4}{7 + 2 \cos \theta}$

Identify the conic with a focus at the origin, and then give the directrix and eccentricity.

$r \left(3 + 5 \sin \theta\right) = 11$

Identify the conic with a focus at the origin, and then give the directrix and eccentricity.

$r \left(7 + 8 \cos \theta\right) = 7$

Convert the polar equation of a conic section to a rectangular equation.

$r = \frac{2}{5 - 3 \sin \theta}$

Convert the polar equation of a conic section to a rectangular equation.

$r = \frac{3}{2 + 5 \cos \theta}$

Convert the polar equation of a conic section to a rectangular equation.

$r = \frac{3}{8 - 8 \cos \theta}$

Convert the polar equation of a conic section to a rectangular equation.

$r = \frac{5}{5 - 11 \sin \theta}$

Convert the polar equation of a conic section to a rectangular equation.

$r \left(2 - \cos \theta\right) = 1$

Convert the polar equation of a conic section to a rectangular equation.

$r = \frac{6 \sec \theta}{- 2 + 3 \sec \theta}$

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.

Directrix: $x = 4 ; e = \frac{1}{5}$

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.

Directrix: $y = 2 ; e = 2$

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.

Directrix: $x = 1 ; e = 1$

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.

Directrix: $x = - \frac{1}{4} ; e = \frac{7}{2}$

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.

Directrix: $y = 4 ; e = \frac{3}{2}$

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.

Directrix: $x = −5 ; e = \frac{3}{4}$

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.

Directrix: $x = −3 ; e = \frac{1}{3}$

Recall from Rotation of Axes that equations of conics with an $x y$ term have rotated graphs. For the following exercises, express each equation in polar form with $r$ as a function of $\theta .$

$x^{2} + x y + y^{2} = 4$

Recall from Rotation of Axes that equations of conics with an $x y$ term have rotated graphs. For the following exercises, express each equation in polar form with $r$ as a function of $\theta .$

$16 x^{2} + 24 x y + 9 y^{2} = 4$

Write the equation of the ellipse in standard form. Then identify the center, vertices, and foci.

$\frac{x^{2}}{25} + \frac{y^{2}}{64} = 1$

Write the equation of the ellipse in standard form. Then identify the center, vertices, and foci.

$9 x^{2} + y^{2} + 54 x - 4 y + 76 = 0$

Use the given information to find the equation for the ellipse.

Center at $\left(0 , 0\right) ,$ focus at $\left(3 , 0\right) ,$ vertex at $\left(−5 , 0\right)$

Use the given information to find the equation for the ellipse.

A whispering gallery is to be constructed such that the foci are located 35 feet from the center. If the length of the gallery is to be 100 feet, what should the height of the ceiling be?

Write the equation of the hyperbola in standard form. Then give the center, vertices, and foci.

$\frac{\left(y + 1\right)^{2}}{16} - \frac{\left(x - 4\right)^{2}}{36} = 1$

Write the equation of the hyperbola in standard form. Then give the center, vertices, and foci.

$3 x^{2} - y^{2} - 12 x - 6 y - 9 = 0$

Find the equation of the hyperbola.

Foci at $\left(3 , 7\right)$ and $\left(7 , 7\right) ,$ vertex at $\left(6 , 7\right)$

Write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.

$\left(x + 2\right)^{2} = \frac{1}{2} \left(y - 1\right)$

Write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.

$x^{2} + 10 x - y + 23 = 0$

Write the equation of the parabola using the given information.

Focus at $\left(2 , \frac{9}{8}\right) ;$ directrix is $y = \frac{7}{8}$

Determine which of the conic sections is represented.

$16 x^{2} + 24 x y + 9 y^{2} + 24 x - 60 y - 60 = 0$

Determine which of the conic sections is represented.

$4 x^{2} + x y + 2 y^{2} + 8 x - 26 y + 9 = 0$

Determine the angle $\theta$ that will eliminate the $x y$ term, and write the corresponding equation without the $x y$ term.

$x^{2} - x y + y^{2} - 6 = 0$

Given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix.

$r = \frac{10}{1 - 5 \cos \theta}$

Given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix.

$r = \frac{1}{4 + 3 \sin \theta}$

Given information about the graph of a conic with focus at the origin, find the equation in polar form.

Directrix is $x = 3$ and eccentricity $e = 1$

Write the equation in standard form and state the center, vertices, and foci.

$\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$

Sketch the graph, identifying the center, vertices, and foci.

Write the standard form equation of an ellipse with a center at $\left(1 , 2\right) ,$ vertex at $\left(7 , 2\right) ,$ and focus at $\left(\right. 4 , 2 ).$

Write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes.

$\frac{x^{2}}{49} - \frac{y^{2}}{81} = 1$

Graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes.

Write the standard form equation of a hyperbola with foci at $\left(1 , 0\right)$ and $\left(1 , 6\right) ,$ and a vertex at $\left(1 , 2\right) .$

Write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix.

$3 x^{2} - 12 x - y + 11 = 0$

Graph the parabola, labeling the vertex, focus, and directrix.

A searchlight is shaped like a paraboloid of revolution. If the light source is located 1.5 feet from the base along the axis of symmetry, and the depth of the searchlight is 3 feet, what should the width of the opening be?

Determine which conic section is represented by the given equation, and then determine the angle $\theta$ that will eliminate the $x y$ term.

$x^{2} + 4 x y + 4 y^{2} + 6 x - 8 y = 0$

Identify the conic with focus at the origin, and then give the directrix and eccentricity.

$r = \frac{5}{4 + 6 \cos \theta}$