Question

Identify the type of conic section given by the polar equation below. Also give the equation of its directrix (in rectangular coordinates is fine.)

r = 8/4+ cos θ.

Answer 1

x = ±8

Explanation:

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=(ep)/(1 \pm e*cos\theta)\\ \\`

From the question, the polar equation of the circle is:

`r=(8)/(4+cos\theta)`

We have to make the equation to be in the form of
`r=(ep)/(1 \pm e*cos\theta)\\ \\`. Therefore:

`r=(8)/(4+cos\theta)\\\\Multiply \ through\ numerator\ and\ denminator\ by\ (1)/(4)\\\\ r=(8*(1)/(4) )/((4+cos\theta)*(1)/(4) )\\\\r=(2)/(4*(1)/(4) +cos\theta*(1)/(4))\\ \\r=((1)/(4)*8)/(1+(1)/(4)cos\theta)`

This means that the eccentricity (e) = 1/4 and the equation of the directrix is x = ±8