polar equation

**calcbeg** · November 14th 2009, 02:13 PM

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

**Soroban** · November 14th 2009, 02:34 PM

Hello, calcbeg!

I must assume you know these conversions:

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

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

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

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

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

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

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

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

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