Problem source

\noindent \begin{center}
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\par\end{center}

In the diagram, $O$ is the origin, $P$ is a point of a curve $r=r(\theta)$
with coordinates $(r,\theta)$ and $Q$ is another point of the curve,
close to $P$, with coordinates $(r+\delta r,\theta+\delta\theta).$
The angle $\angle PRQ$ is a right angle. By calculating $\tan\angle QPR,$
show that the angle at which the curve cuts $OP$ is 
\[
\tan^{-1}\left({\displaystyle r\dfrac{\mathrm{d}\theta}{\mathrm{d}r}}\right).
\]

Let $\alpha$ be a constant angle, $0<\alpha<\frac{1}{2}\pi$. The
curve with the equation 
\[
r=\mathrm{e}^{\theta\cot\alpha}
\]
in polar coordinates is called an \textit{equiangular spiral}. Show
that it cuts every radius line at an angle $\alpha.$ Sketch the spiral. 

Find the length of the complete turn of the spiral beginning at $r=1$
and going outwards. What is the total length of the part of the spiral
for which $r\leqslant1$? 

{[}You may assume that the arc length $s$ of the curve satisfies
\[
{\displaystyle \left(\frac{\mathrm{d}s}{\mathrm{d}\theta}\right)^{2}=r^{2}+\left(\frac{\mathrm{d}r}{\mathrm{d}\theta}\right)^{2}.}]
\]