Question

question 29 · 1 point identify the conic that the following polar equation represents. $r = \frac{8}{2 + 2\sin\theta}$ select the correct answer below: parabola ellipse hyperbola

Explanation:

Step1: Rewrite the polar equation

First, rewrite $r=\frac{8}{2 + 2\sin\theta}$ as $r=\frac{4}{1+\sin\theta}$. The general form of a polar - conic equation is $r=\frac{ed}{1 + e\sin\theta}$, where $e$ is the eccentricity and $d$ is the distance from the focus to the directrix.

Step2: Identify the eccentricity

Comparing $r=\frac{4}{1+\sin\theta}$ with $r=\frac{ed}{1 + e\sin\theta}$, we can see that $e = 1$.

Step3: Determine the conic

If $e = 1$, the conic is a parabola. If $e<1$, it is an ellipse, and if $e>1$, it is a hyperbola. Since $e = 1$ for the given equation, the conic is a parabola.

Answer:

A. Parabola