A hyperbola is the set of all points in a plane where the absolute difference of the distances from two fixed points (foci) is constant.
Let \( F \) and \( F_1 \) be the foci. Then:
\[ FM - F_1M = AA_1 = 2a, \quad FF_1 = 2c, \quad c^2 - a^2 = b^2 \]
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] This represents a hyperbola that opens left and right.
\[ \varepsilon = \frac{c}{a} = \frac{\sqrt{a^2 + b^2}}{a}, \quad \varepsilon > 1 \]
For a point \( M \) on the hyperbola:
\[ r = \varepsilon x - a, \quad r_1 = \varepsilon x + a \]
