# How do you find the eccentricity, directrix, focus and classify the conic section r=14.4/(2-4.8costheta)?

Sep 19, 2016

See below.

#### Explanation:

We have

$r = \frac{a}{b - c \cos \theta}$ or

$\sqrt{{x}^{2} + {y}^{2}} = \frac{a}{b - c \frac{x}{\sqrt{{x}^{2} + {y}^{2}}}}$ or

$\sqrt{{x}^{2} + {y}^{2}} = \frac{a \sqrt{{x}^{2} + {y}^{2}}}{b \sqrt{{x}^{2} + {y}^{2}} - c x}$ or

$1 = \frac{a}{b \sqrt{{x}^{2} + {y}^{2}} - c x}$ or
$b \sqrt{{x}^{2} + {y}^{2}} - c x = a$ or

${x}^{2} + {y}^{2} = {\left(c x + a\right)}^{2} / {b}^{2}$ or

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

so we have supposing $b \ne 0$

$c = b$ parabola
$c > b$ hyperbola
$c < b$ ellipse
$c = 0$ circle