Solve, a. If necessary, write the equation in one of the standard forms for a...

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: $r=\frac{ep}{1+esin\theta }$ Now, we will divide the numerator and denominator by 2. $r=\frac{\frac{6}{2}}{\frac{2}{2}+\frac{1}{2}sin\theta }$
$=3/(1+1/2sin\theta )$
Hence, the required standard form is $r=\frac{3}{1+\frac{1}{2}sin\theta }$ (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 $e=\frac{1}{6}andp=6$. (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.