Hyperbola and its Types

A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant.

The equation of a hyperbola is simplest if the centre of the hyperbola is at the origin and the foci are on the x-axis or y-axis.  The two such possible orientations in following fig.

Let `F_1` and `F_2` be the foci and O be the mid-point of the line segment 
`F_1F_2`.  Let O be the origin and the line through O through `F_2` be the positive x-axis and that through `F_1` as the negative x-axis.  The line through O perpendicular to the x-axis be the y-axis.  Let the coordinates of `F_1` be (– c,0) and `F_2` be (c,0) in following fig.

Let P(x, y) be any point on the hyperbola such that the difference of the distances from P to the farther point minus the closer point be 2a. So given,  `PF_1 – PF_2 = 2a`
Using the distance formula, we have
`sqrt((x + c)^2+y^2) - sqrt((x-c)^2 + y^2) =2a`

i.e., `sqrt((x + c)^2+y^2) = 2a + sqrt((x-c)^2 + y^2) `

Squaring both side, we get

`(x + c)^2 + y^2 = 4a^2 + 4a  sqrt((x-c)^2 + y^2) + ((x + c)^2+y^2)`

and on simplifying, we get

`(cx)/a-a = sqrt((x-c)^2 + y^2)`

On squaring again and further simplifying, we get

`x^2/a^2 - y^2/(c^2 -a^2) = 1
i.e., x^2/a^2 -y^2/b^2=1`                       (since `c^2-a^2=b^2`)

Hence any point on the hyperbola satisfies  `x^2/a^2 -y^2/b^2=1`             

From the standard equations of hyperbolas, we observ that:

1. Hyperbola is symmetric with respect to both the axes, since if (x, y) is a point on the hyperbola, then (– x, y), (x, – y) and (– x, – y) are also points on the hyperbola.

2. The foci are always on the transverse axis.  It is the positive term whose denominator gives the transverse axis. 

The term “difference” that is used in the definition means the distance to the farther point  minus the distance to the closer point. The two fixed points are called the foci of the hyperbola. The mid-point of the line segment joining the foci is called the centre of the hyperbola. The line through the foci is called the transverse axis and the line through the centre 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 in the above fig.

The distance between the two foci by 2c, the distance between two vertices (the length of the transverse axis) by 2a and we define the quantity b as b = `sqrt(c^2-a^2)`
Also 2b is the length of the conjugate axis in above fig.
To find the constant `P_1F_2 – P_1F_1` :

By taking the point P at A and  B   in the  above Fig,  we have
`BF_1  – BF_2 =  AF_2 – AF_1`
(by the definition of the hyperbola) 
`BA +AF_1– BF_2 = AB + BF_2– AF_1` 
i.e., `AF_1 =  BF_2` 
So that,  `BF_1 – BF_2 =  BA + AF_1– BF_2 = BA = 2a`