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Thread: polar equation

  1. #1
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    polar equation

    Hi

    Show that the polar equation r= a cos x + b sin x where ab not equal 0 represents a circle and find its center and radius

    I really have no idea where to start - any ideas at all??

    Thanks

    Struggling calculus beginner
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  2. #2
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    Hello, calcbeg!

    I must assume you know these conversions:

    . . $\displaystyle \begin{array}{ccc}r\cos\theta &=& x \\ r\sin\theta &=& y\\ r^2 &=& x^2 + y^2\end{array}$


    Show that the polar equation: .$\displaystyle r \:=\:a\cos\theta + b\sin\theta,$ .where $\displaystyle ab \neq 0,$
    represents a circle and find its center and radius.

    We have: .$\displaystyle r \;=\;a\cos\theta + b\sin\theta $

    $\displaystyle \text{Multiply by }r\!:\;\;\underbrace{r^2}_{x^2+y^2} \;=\;a\underbrace{r\cos\theta}_x + b\underbrace{r\sin\theta}_y $

    So we have: .$\displaystyle x^2 + y^2 \;=\;ax + by \quad\Rightarrow\quad x^2 - ax + y^2 - by \;=\;0$

    Complete the square: .$\displaystyle x^2 - ax + \frac{a^2}{4} + y^2 - by + \frac{b^2}{4} \;\;=\;\;0 + \frac{a^2}{4} + \frac{b^2}{4} $

    Simplify: .$\displaystyle \left(x - \frac{a}{2}\right)^2 + \left(y - \frac{b}{2}\right)^2 \;=\;\frac{a^2+b^2}{4}$


    This is the equation of a circle: .$\displaystyle \begin{Bmatrix}\text{Center:} & \left(\dfrac{a}{2},\:\dfrac{b}{2}\right) \\ \\[-3mm] \text{Radius:} & \dfrac{\sqrt{a^2+b^2}}{2} \end{Bmatrix}$

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