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Thread: polar coordinates

  1. #1
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    polar coordinates

    A circle C has center at the origin and radius 3. Another circle K has a diameter with one end at the origin and the other end at the point (0, 13). The circles C and K intersect in two points. Let P be the point of intersection of C and K which lies in the first quadrant. Let $\displaystyle (r,\theta) $ be the polar coordinates of P, chosen so that $\displaystyle r $is positive and $\displaystyle 0 \leq \theta \leq \pi/2$

    Find $\displaystyle r $ and $\displaystyle \theta$
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  2. #2
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    Quote Originally Posted by viet View Post
    A circle C has center at the origin and radius 3. Another circle K has a diameter with one end at the origin and the other end at the point (0, 13). The circles C and K intersect in two points. Let P be the point of intersection of C and K which lies in the first quadrant. Let $\displaystyle (r,\theta) $ be the polar coordinates of P, chosen so that $\displaystyle r $is positive and $\displaystyle 0 \leq \theta \leq \pi/2$

    Find $\displaystyle r $ and $\displaystyle \theta$
    So you are looking for the intersections of the circles
    $\displaystyle x^2 + y^2 = 9$
    and
    $\displaystyle x^2 + \left (y - \frac{13}{2} \right )^2 = \left ( \frac{13}{2} \right ) ^2$

    From the first equation $\displaystyle x^2 = 9 - y^2$. Inserting this into the second equation gives:
    $\displaystyle (9 - y^2) + \left (y - \frac{13}{2} \right ) ^2 = \frac{169}{4}$

    Solve this for y. I get:
    $\displaystyle y = \frac{9}{13}$

    There are two x values for this y value:
    $\displaystyle x = \pm \sqrt{9 - \left ( \frac{9}{13} \right ) ^2} = \pm \frac{12}{13} \sqrt{10}$

    Thus the point of intersection in QI is
    $\displaystyle \left ( \frac{12}{13} \sqrt{10}, \frac{9}{13} \right )$

    I leave putting this into polar coordinates to you.

    -Dan
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  3. #3
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    Hello, viet!

    A circle $\displaystyle C$ has center at the origin and radius 3.
    Another circle $\displaystyle K$ has a diameter with one end at the origin and the other end at the point (0, 13).
    The circles $\displaystyle C$ and $\displaystyle K$ intersect in two points.

    Let $\displaystyle P$ be the point of intersection of $\displaystyle C$ and $\displaystyle K$ which lies in the first quadrant.
    Let $\displaystyle (r,\,\theta) $ be the polar coordinates of $\displaystyle P$, chosen so that $\displaystyle r$ is positive and $\displaystyle 0 \leq \theta \leq \pi/2$

    Find $\displaystyle r$ and $\displaystyle \theta$
    If you are familiar with polar equations of circle, it's easier.


    Circle $\displaystyle C$ has the equation: .$\displaystyle r\:=\:3$

    Circle $\displaystyle K$ has the equation: .$\displaystyle r \:=\:13\sin\theta$

    They intersect where: .$\displaystyle 13\sin\theta \:=\:3$

    Hence, we have: .$\displaystyle \sin\theta \:=\:\frac{3}{13}\quad\Rightarrow\quad\theta \:=\:\sin^{-1}\left(\frac{3}{13}\right) \:=\:0.232868178 \:\approx\:0.233$


    Therefore, point $\displaystyle P$ is: .$\displaystyle 3,\,0.233)$

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