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Thread: Polar Equations

  1. #1
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    Polar Equations

    Given the polar equation r=4/(2+2sin(x)).Find the Directrix,eccentricity and sketch the conic.
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  2. #2
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    Hello, Gracy!

    You're expected to know the general form for polar conics.

    . . $\displaystyle r \:=\:\frac{ke}{1 + e\sin\theta}\qquad\text{or}\qquad r\:=\:\frac{ke}{1 + e\cos\theta}$ . where $\displaystyle e$ = eccentricity

    And these rules: .$\displaystyle \begin{array}{ccc} e < 1 & \text{ellipse} \\ e = 1 & \text{parabola} \\ e > 1 & \text{hyperbola}\end{array}$


    Given the polar equation: .$\displaystyle r\:=\:\frac{4}{2+2\sin x}$
    Find the directrix, eccentricity and sketch the conic.

    We have: .$\displaystyle r \:=\:\frac{2}{1 + \sin x}\quad\Rightarrow\quad\boxed{e = 1}$

    Since $\displaystyle e = 1$, we have a parabola with its focus is at the pole (origin).

    Plot a few points: .$\displaystyle \begin{array}{ccc}x = 0 & r = 2 \\ x = \frac{\pi}{2} & r = 1 \\ x = \pi & r = 2\end{array}$

    The graph looks like this:
    Code:
                         2|
              - - - - - - + - - - - - -
                          |
                         1|
                         ***
                    *     |     *
                 *        |        *
          ------*---------+---------*------
                          |
               *          |          *
                          |

    The focus is 1 unit from the vertex.
    . . Hence, the directrix is also 1 unit from the vertex.
    It is the horizontal line: $\displaystyle y \:=\:2$
    . . Converting to polars: .$\displaystyle r\sin x = 2\quad\Rightarrow\quad\boxed{ r = 2\csc x}$

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