Find a polar equation of the conic with the given characteristics and with one focus at the pole.Conic Eccentricity DirectrixHyperbola

Question

Quizplus · Accepted Answer

Since the hyperbola has eccentricity greater than 1, its polar equation will be of the form $r = \frac{ed}{1\pm e\cos	heta}$, where $e$ is the eccentricity and $d$ is the distance from the pole to the directrix. Since one focus is at the pole, the distance from the focus to the directrix is also $d$, and we can use the formula $c=\sqrt{a^2+b^2}$, where $a$ is the distance from the center to a vertex and $b$ is the distance from the center to a co-vertex, to find that $d=\sqrt{c^2-a^2}=3\sqrt{3}$ (note that $a=b=\sqrt{10}$). Thus, the polar equation is $r = \frac{\sqrt{39}}{1\pm \frac{\sqrt{39}}{3}\cos	heta}$, which matches choice B.

Find a Polar Equation of the Conic with the Given