Foci of a hyperbola

By John
2 Min Read

A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is constant. ‘Difference’ means the distance to the ‘farther’ point minus the distance to the ‘closer’ point. The two fixed points are the foci and the mid-point of the line segment joining the foci is the center of the hyperbola.

The line through the foci is called the transverse axis. Also, the line through the center and perpendicular to the transverse axis is called the conjugate axis. The points at which the hyperbola intersects the transverse axis are called the vertices of the hyperbola.

The distance between the two foci is: 2c
The distance between two vertices is: 2a (this is also the length of the transverse axis)
The length of the conjugate axis is 2b … where b = √ (c2 – a2)
Finding the Constant P1F2 – P1F1
Refer the diagram below:

We take a point P at A and B as shown above. Therefore, by the definition of a hyperbola, we have

BF1 – BF2 = AF2 – AF1
∴ BA + AF1 – BF2 = AB + BF2 – AF1

Solving the equation, we get, AF1 = BF2
Hence, BF1 – BF2 = BA + AF1 – BF2 = BA = 2a.