Mathematics

For a hyperbola, the key is understanding how the two axes relate through the foci. The distance from center to focus is $c$, and unlike an ellipse, the relationship here is $c^2 = a^2 + b^2$ (not subtraction). This comes from the hyperbola's defining property: the difference of distances to the two foci is constant, which geometrically forces the focal distance to exceed the semi-major axis.

For a hyperbola, the key is understanding how the two axes relate through the foci. The distance from center to focus is $c$, and unlike an ellipse, the relationship here is $c^2 = a^2 + b^2$ (not subtraction). This comes from the hyperbola's defining property: the difference of distances to the two foci is constant, which geometrically forces the focal distance to exceed the semi-major axis.