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.

Question
Conic sections
asked 2020-11-22
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.

Answers (1)

2020-11-23
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. \(r = \frac{\frac{6}{2}}{\frac{2}{2} + \frac{1}{2} sin \theta}\)
\(= 3/(1 + 1/2 sin \theta)\)
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 \(e= \frac{1}{6} and p = 6\). (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.
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