Solve, a. If necessary, write the equation in one of the standard forms for a conic in polar coordinates r=62+sinθ b. Determine values for e and p. Use the value of e to identify the conic section. c. Graph the given polar equation.

Question

Solve,  a. If necessary, write the equation in one of the standard forms for a conic in polar coordinates r=62+sinθ  b. Determine values for e and p. Use the value of e to identify the conic section.  c. Graph the given polar equation.

Leonard Stokes · Accepted Answer

We will first consider the given polar coordinates, r=62+sinθ(a) The objective is to write the equation in standard form. Since we know that the standard form is as follows: r=ep1+esinθ Now, we will divide the numerator and denominator by 2. r=6222+12sinθ  =3/(1+1/2sinθ) Hence, the required standard form is r=31+12sinθ (b) The next objective is to determine the values of e and p. On comparing with standard form, we get, e=12 ep=3⇒p=6 Thus, the values are e=16andp=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=12. Hence, we can conclude that the given conic equation is of ellipse.

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.

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