The distance from P to F is given by PF = x − c 2 + y − 0 2 = x − c 2 + y 2.
The distance from a general point P x, y to E is given by PE = x − − c 2 + y − 0 2 = x + c 2 + y 2. So, the distance from each focus to the center is c. The general equation for an "East-West opening hyperbola" is: x − h 2 a 2 − y − k 2 b 2 = 1, while the general equation for a "North-South opening hyperbola" is: y − k 2 a 2 − x − h 2 b 2 = 1, where h, k is the center, a is the length of the semi-major axis (the distance from each vertex to the center), and b is the length of the semi-minor axis (the perpendicular distance from each vertex to each asymptote).ĭerivation of the general equation from the focal propertyįor simplicity, let's say that the hyperbola is centered at 0, 0 with the following foci: E at − c, 0 and F at c, 0. In particular, the difference between these distances is always equal to the length of the major axis. Ī hyperbola can be described as the locus of points for which the absolute value of the difference between the distances from any point P to each focus is a constant. Beyond the vertices on the same line as the major axis (lying further from the center) there are two points, E and F, known as the foci. The line segment between the two vertices is known as the transverse or major axis. The points on these branches which are closest together, and thus closest to the center, are called vertices. A hyperbola consists of two open, disconnected curves called branches, which are mirror images of each other and resemble infinite bows.