Find a polar equation of the conic with its focus at the pole.
Conics
Eccentrity
Directrix
Hyperbola

Question

Find a polar equation of the conic with its focus at the pole. ​
Conics
​Eccentrity
Directrix
Hyperbola   ​

Quizplus · Accepted Answer

Since the focus is at the pole, the polar equation will have the form $r=e\cdot\frac{d}{1+e\cos	heta}$, where $d$ is the distance from the pole to the directrix and $e$ is the eccentricity.

Since we have a hyperbola, we know that $e>1$. Also, we know the eccentricity is $e=2$, since the distance between the foci is $2ae$ and one of the foci is at the pole.

Finally, we need to find $d$, the distance from the pole to the directrix. Since one of the foci is at the pole, we know that the distance from the pole to the center is $c=\frac{a}{e}=\frac12$.

Since this is a hyperbola whose distance between the foci is $10$, we know that $2a=10$, so $a=5$.

The distance from the center to the directrix is $d=a/e=5/2$.

Putting it all together, we get the polar equation $\boxed{r=10/(1+2\cos	heta)}$.

Find a Polar Equation of the Conic with Its Focus