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I have a curve C with polar equation $r^2 = a^2\cos{2\theta} $ enter image description here

And I am looking to find the length $x$ when $r=max$

Judging from the equation: $r = \sqrt{a^2\cos{2\theta}} $

R will be maximum at $\cos{2\theta}=1$

So the maximum value of $r$ is:

$r = \sqrt{a^2} =a$

However the derivative disagrees as: $x^2=r^2\sin^2{\theta}=a^2\cos{2\theta}\,\sin^2{\theta} \\ \frac{d}{d\theta}\left (a^2\cos{2\theta}\,\sin^2{\theta} \right )=a^2(2\sin{\theta}\cos{\theta}-8\sin^3{\theta}\cos{\theta}) \\ \sin{\theta}=\frac{1}{2} \\ \theta= \frac{\pi}{6} \\ r= \frac{a}{\sqrt{2}}$

What am I doing wrong?

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    Ohh, I mixed up $r$ and $y$ right there. So $\frac{d}{d\theta}\left (a^2 \cos{2 \theta} \cos^2{\theta} \right )=0$ will end up giving me $r=a$ right?2012-06-21

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Differentiating $r$ wrt to $\theta$ gives $\theta \mapsto |a|\frac{\sin 2 \theta}{\cos 2 \theta}$, which is zero when $\cos 2 \theta = \pm 1$. No disagreement there!

From this compute $x = r \cos \theta$. Since $\theta = 0$ maximizes $r$, the corresponding $x = |a|$.