A hyperbola balances two focal distances by keeping their difference constant.
WHAT HAPPENS IF…Is the sum or the difference of focal distances constant?
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Connect the picture to the ideaa2x2−b2y2=1,c2=a2+b2
The identity cosh²t−sinh²t=1 verifies the parametrization (a cosh t,b sinh t). Its foci are (±c,0), and the absolute difference of distances is 2a. The two branches correspond to which focus is farther away. The asymptotes are y=±(b/a)x.