1. ## Polar Coordinates

I need help on this second part to this question...

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Q: Sketch the curve with the equation $r=a(1+ \cos\theta)$ for $0 \le \theta \le \pi$ where $a>0$. Sketch also the line with the equation $r=2a \sec \theta$ for $- \frac{\pi}{2} < \theta < \frac{\pi}{2}$ on the same diagram.

The half-line with the equation $\theta = \alpha$, $0<\alpha<\frac{\pi}{2}$, meets the curve at $A$ and the line with equation $r=2a \sec \theta$ at $B$. If $O$ is the pole, find the value of $\cos \alpha$ for which $OB=2OA$.

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I've drawn the graph but can't do the second part to the question. Can someone help please? Thank you in advance.

2. Originally Posted by Air
I need help on this second part to this question...

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Q: Sketch the curve with the equation $r=a(1+ \cos\theta)$ for $0 \le \theta \le \pi$ where $a>0$. Sketch also the line with the equation $r=2a \sec \theta$ for $- \frac{\pi}{2} < \theta < \frac{\pi}{2}$ on the same diagram.

The half-line with the equation $\theta = \alpha$, $0<\alpha<\frac{\pi}{2}$, meets the curve at $A$ and the line with equation $r=2a \sec \theta$ at $B$. If $O$ is the pole, find the value of $\cos \alpha$ for which $OB=2OA$.

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I've drawn the graph but can't do the second part to the question. Can someone help please? Thank you in advance.
Well, to find OA you need to find the points where the curve and half line meet. So we would have
$r = a(1 + cos(\theta)) = a(1 + cos(\alpha))$

So the coordinates for the point A are $(r, \alpha)$. Thus OA is simply $r = (1 + cos(\alpha) )$.

Now you do the same for OB. Then using OB = 2OA you can write an equation for $cos(\alpha)$.

-Dan