Hyperbola can be defined as the locus of point that moves such that the difference of its distances from two fixed points called the foci is constant. The constant difference is the length of the transverse axis, 2a.
From the general equation of any conic (A and C have opposite sign, and can be A > C, A = C, or A
$Ax^2 - Cy^2 + Dx + Ey + F = 0 \,$ or
$\, x^2 - Cy^2 + Dx + Ey + F = 0$