Question

Find the eccentricity, b. identify the conic, c. give an equation of the directrix, and d. sketch the conic.

Answer 1

Answer:

a) 10/3  

b) hyperbola

c) x = ± 6/5

Step-by-step explanation:

a) A conic section with a focus at the origin, a directrix of x = ±p where p is a positive real number and positive eccentricity (e) has a polar equation:

$r=\frac{ep}{1\pm e*cos\theta}$

Given the conic equation: $r=\frac{12}{3-10cos\theta}$

We have to make it to be in the form $r=\frac{ep}{1\pm e*cos\theta}$:

$r=\frac{12}{3-10cos\theta}\\\\multiply\ both\ sides\ by\ \frac{1}{3} \\\\r=\frac{12*\frac{1}{3}}{(3-10cos\theta)*\frac{1}{3}}\\\\r=\frac{12*\frac{1}{3}}{3*\frac{1}{3}-10cos\theta*\frac{1}{3}}\\\\r=\frac{4}{1-\frac{10}{3}cos\theta } \\\\r=\frac{\frac{10}{3}(\frac{6}{5} ) }{1-\frac{10}{3}cos\theta }$

Comparing with  $r=\frac{ep}{1\pm e*cos\theta}$

e = 10/3 = 3.3333, p = 6/5

b) since the eccentricity = 3.33 > 1, it is a hyperbola

c) The equation of the directrix is x = ±p = ± 6/5