Let $P$ be any point and let $OP=r$, where $O$ is the origin and let $OP$ make an angle $\theta$ with the axis, then $r$ and $\theta$ are called Polar coordinates of the point $P$, and the coordinates may be written as $(r,\theta)$.

Find the Cartesian equations of

(b) $r\cos(\theta-\alpha)=p$

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$\cos(\theta-\alpha)=p$

$\cos(\theta-\alpha)$ may be expanded

$\therefore r\cos\theta\cos\alpha+r\sin\theta\sin\alpha=p$

Therefore the Cartesian equation of $r\cos(\theta-\alpha)=p$ is

$x\cos\alpha+y\sin\alpha=p$

$Note$. The perpendicular from the origin to this line is of length $p$ and makes an angle $\alpha$ with the x-axis. This form of equation of a straight line is known as the normal or perpendicular form.

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