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Q Find the curve whose polar subtangent is constant

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I have done this thing::
[tex]\frac{rsin(\theta)}{k}= - \frac{r+tan(\theta) \frac{dr}{d\theta}}{-rtan\theta + \frac{dr}{d\theta}}[/tex]

Rearranging and solving i got

[tex]rsin \theta \frac{d(rcos\theta)}{d\theta}=-k\frac{d(rsin\theta)}{d\theta}[/tex]

on further solving i got
[tex] rcos\theta= -kln(rsin\theta) +c [/tex]

But the answer at the back is ::
:: The Reciprocal Spiral ::
:: r(\theta - \alpha) = c::

Where I went wrong ::Pls Help
 

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