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 ?
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
and the above already proves $\;(1)\;$ . Now doing a little analytic geometry:
and we have the standard equation of a hyperbola.
Try now to complete your question's answer.