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In this exercise it is required to verify that the points O & A(4,0) are the vertices of the Hyperbola H , as you can see in the marked part.

Can someone help verify that ?

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Some ideas:

If $\,M=(h,k)\;$ , then it must be that $\,MM'=|h-1|=r\;,\;r=$ the circle's radius.

We also have, since the triangle $\,\Delta TFM\;$ is a $\,30^\circ-60^\circ-90^\circ\;$ triangle, that

$$MF=2r\;,\;FT=\sqrt 3\,r$$

and the above already proves $\;(1)\;$ . Now doing a little analytic geometry:

$$4=\frac{MF^2}{MM'^2}=\frac{(h+2)^2+k^2}{r^2}\implies (h+2)^2+k^2=4(h-1)^2\implies$$

$$h^2+k^2+4h+4=4h^2-8h+4\implies 3(h^2-4h)-k^2=0\implies$$

$$3\left(h-2\right)^2-k^2=12\implies \frac{(h-2)^2}{2^2}-\frac{k^2}{(2\sqrt3)^2}=1$$

and we have the standard equation of a hyperbola.

Try now to complete your question's answer.

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