Question

# Solve, a. If necessary, write the equation in one of the standard forms for a conic in polar coordinates r = frac{6}{2 + sin theta} b. Determine values for e and p. Use the value of e to identify the conic section. c. Graph the given polar equation.

Conic sections
Solve, a. If necessary, write the equation in one of the standard forms for a conic in polar coordinates $$r = \frac{6}{2 + sin \theta}$$ b. Determine values for e and p. Use the value of e to identify the conic section. c. Graph the given polar equation.

We will first consider the given polar coordinates, $$r =\frac{6}{2 + sin \theta} (a)$$ The objective is to write the equation in standard form. Since we know that the standard form is as follows: $$r = \frac{ep}{1 + e sin \theta}$$ Now, we will divide the numerator and denominator by 2. $$r = \frac{\frac{6}{2}}{\frac{2}{2} + \frac{1}{2} sin \theta}$$
$$= 3/(1 + 1/2 sin \theta)$$
$$r = \frac{3}{1 + \frac{1}{2} sin \theta}$$ (b) The next objective is to determine the values of e and p. On comparing with standard form, we get, $$e = \frac{1}{2}$$ $$ep = 3 \Rightarrow p = 6$$ Thus, the values are $$e= \frac{1}{6} and p = 6$$. (c) Next, identify the conic section using the value of eccentricity. Since we know that the eccentricity of an ellipse which is not a circle is greater than zero but less than 1. Here, $$e = \frac{1}{2}$$. Hence, we can conclude that the given conic equation is of ellipse.