# Hyperbola

Here's a visualization of a hyperbola with a horizontal transverse axis centered around the origin. Its formula is *x*^{2}/*a*^{2} − *y*^{2}/*b*^{2} = 1. Using the sliders you can alter *a* and *b*, the distances from the origin to each vertex or covertex respectively. (Or, put another way, 2*a* is the length of the transverse axis and 2*b* is the length of the conjugate axis.) *c* is defined in the usual Pythagorean manner: *a*^{2}+*b*^{2} = *c*^{2}.

The diagonals of the 2*a* × 2*b* rectangle centered on the origin form the asymptotes. Do you see why the three segments labeled *c* are the same length? Do you see why *a*^{2}+*b*^{2} = *c*^{2}? The circle centered on the origin may help.

The defining characteristic of a hyperbola is that the distances between a point *P* on the hyperbola and the two foci differ by a constant. In this case, the constant difference is 2*a*. Look at the circle centered on point *P* and the circle centered on the focus *F*_{1}. Circle *F*_{1} always has radius 2*a* regardless of the position of *P*, representing the constant difference. Circle *P* has a radius that varies as *P* moves, but its radius is always the distance between *P* and *F*_{2}. Thus you can see the constant difference between the distances *PF*_{1} and *PF*_{2}.

Try it! Press the "Open Geogebra" button; then adjust the *a* and *b* sliders and see what happens. Also move point *P* along the hyperbola.

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