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Convert the conic $\text{\hspace{0.17em}}r=\frac{1}{5-5\mathrm{sin}\text{\hspace{0.17em}}\theta}$ to rectangular form.
We will rearrange the formula to use the identities $r=\sqrt{{x}^{2}+{y}^{2}},x=r\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta ,\text{and}y=r\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta .$
Convert the conic $\text{\hspace{0.17em}}r=\frac{2}{1+2\text{}\mathrm{cos}\text{}\theta}\text{\hspace{0.17em}}$ to rectangular form.
$4-8x+3{x}^{2}-{y}^{2}=0$
Access these online resources for additional instruction and practice with conics in polar coordinates.
Visit this website for additional practice questions from Learningpod.
Explain how eccentricity determines which conic section is given.
If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a hyperbola.
If a conic section is written as a polar equation, what must be true of the denominator?
If a conic section is written as a polar equation, and the denominator involves $\text{\hspace{0.17em}}\mathrm{sin}\text{}\theta ,$ what conclusion can be drawn about the directrix?
The directrix will be parallel to the polar axis.
If the directrix of a conic section is perpendicular to the polar axis, what do we know about the equation of the graph?
What do we know about the focus/foci of a conic section if it is written as a polar equation?
One of the foci will be located at the origin.
For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.
$r=\frac{6}{1-2\text{}\mathrm{cos}\text{}\theta}$
$r=\frac{3}{4-4\text{}\mathrm{sin}\text{}\theta}$
Parabola with $\text{\hspace{0.17em}}e=1\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}\frac{3}{4}\text{\hspace{0.17em}}$ units below the pole.
$r=\frac{8}{4-3\text{}\mathrm{cos}\text{}\theta}$
$r=\frac{5}{1+2\text{}\mathrm{sin}\text{}\theta}$
Hyperbola with $\text{\hspace{0.17em}}e=2\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}\frac{5}{2}\text{\hspace{0.17em}}$ units above the pole.
$r=\frac{16}{4+3\text{}\mathrm{cos}\text{}\theta}$
$r=\frac{3}{10+10\text{}\mathrm{cos}\text{}\theta}$
Parabola with $\text{\hspace{0.17em}}e=1\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}\frac{3}{10}\text{\hspace{0.17em}}$ units to the right of the pole.
$r=\frac{2}{1-\mathrm{cos}\text{}\theta}$
$r=\frac{4}{7+2\text{}\mathrm{cos}\text{}\theta}$
Ellipse with $\text{\hspace{0.17em}}e=\frac{2}{7}\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ units to the right of the pole.
$r(1-\mathrm{cos}\text{}\theta )=3$
$r(3+5\mathrm{sin}\text{}\theta )=11$
Hyperbola with $\text{\hspace{0.17em}}e=\frac{5}{3}\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}\frac{11}{5}\text{\hspace{0.17em}}$ units above the pole.
$r(4-5\mathrm{sin}\text{}\theta )=1$
$r(7+8\mathrm{cos}\text{}\theta )=7$
Hyperbola with $\text{\hspace{0.17em}}e=\frac{8}{7}\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}\frac{7}{8}\text{\hspace{0.17em}}$ units to the right of the pole.
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