Results 1 to 2 of 2

Math Help - Polar Equations

  1. #1
    Junior Member
    Joined
    Oct 2006
    Posts
    71

    Polar Equations

    Given the polar equation r=4/(2+2sin(x)).Find the Directrix,eccentricity and sketch the conic.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,738
    Thanks
    643
    Hello, Gracy!

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

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

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


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

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

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

    Plot a few points: . \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: y \:=\:2
    . . Converting to polars: . r\sin x = 2\quad\Rightarrow\quad\boxed{ r = 2\csc x}

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: May 3rd 2011, 01:16 PM
  2. Replies: 2
    Last Post: April 7th 2010, 02:22 PM
  3. Polar Equations
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: November 16th 2009, 05:51 AM
  4. Replies: 3
    Last Post: June 8th 2009, 06:13 PM
  5. need help with polar equations.
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: May 6th 2009, 03:20 PM

Search Tags


/mathhelpforum @mathhelpforum