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Let A(-α,0) and B(α,0) be two points on the x-axis. A fixed circle with centre at A and radius r lies entirely in a fixed circle with centre at B and radius R. If a moving circle is tangent to the two fixed circles, what's the equation of the locus of the centre of the moving circle?

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  • $\begingroup$ Hint: let $P$ be the center of the moving circle and $r'$ its radius, then $PA,PB$ in terms of $r,R,r'$ are... $\endgroup$ – dxiv Apr 29 '18 at 3:29
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moving circle

Describe the locus in polar co-ordinates centred at the point A

$$r(\theta) = r + x(\theta)$$

where $x(\theta)$ is the radius of the moving circle

find $x(\theta)$ by noting that $x(\theta)= EF = EG$ and applying Cosine Law to the triangle ABE

$|AE| = r+x(\theta) \\ |AB|=2\alpha \\ |BE|= R-x(\theta)$

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  • $\begingroup$ you don't even need to solve for $x(\theta)$ if you know kind of shape you get with the equation $AE + BE = r+R$ $\endgroup$ – mercio Apr 29 '18 at 8:02

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