Hyperbola

 

      The hyperbola is the intersection of double napped cone with plane parallel to the axis.  The hyperbola is the locus of a point in a plane which moves so that its distance from a fixed point bears a constant ratio e (e > 1) to its distance from a fixed line . The fixed point is called the focus S and the fixed line is called the directrix d .

                      

If S is the focus and d is the directrix not containing the focus and P is the moving point, then PS/PM = e , where PM is the perpendicular on thr directrix . e > 1 called eccentricity of the hyperbola  

Standard Eqaution of The Hyperbola :

      Let S be the focus, d be the directrix and e be the eccentricity of a hyperbola . Draw SZ perpendicular to directrix . Let A and A ' divide the segment SZ internally and externally in the ratio e:1. By definition of hyperbola A and A ' lie on hyperbola.



Let AA ' = 2a , midpoint O of segment AA' be the origine . Then O ≡ (0, 0) , A≡ (a, 0) and A ' ≡ (-a, 0)  
 

therefor SA = e AZ.
Let Z ≡ (k, 0)  and S ≡  (h, 0)
By section formula for internal and external division .

a e + a = e k + h ......(2)
-a e + a = e k - h ......(3)
solving these equation , we get
k = a/e   and h = ae
focus S ≡  (ae, 0)  and Z ≡ (a/e, 0)
Equation of the directrix is x = a/e 
That is x - a/e = 0
Let P (x, y) be a point on the hyperbola .
SP = focal distance 
PM = distance of point P from the directrix 

From (1) , (4) and (5)

Squaring both side 

(x - ae)2 + (y - 0)2 = e2x2  - 2 aex  + a2
  x2  - 2 aex + a2e2 + y2 = e2x2 - 2 aex + a2
x2 + a2e2 + y2 = e2x2 + a2 
(1 - e2)x2 + y2  = a2 (1 - e2)
Sience e > 1
(e2  - 1) x2  - y2  = a2 (e2- 1)
 Dividing both side by a2 (e2- 1)
This is the standard eqaution of hyperbola 

Some useful terms of the hyperbola :

i)  The hyperbola intersect X-axis at A(a, 0) a,d A '(-a, 0).
ii) It does not intersect the Y- axis. 
iii) The segment AA ' of length 2a is called the transverse axis and the segment BB ' of length 2b is         called the conjugate axis .
iv) The line segment through the foci of the hyperbola is called the transverse axis and the line segment through centre and perpendicular to transverse axis is conjugate axis. The transverse axis and conjugae axis together are called principal axes of the hyperbola . 
v) Latus rectum is the chord passing through the focus which is perpendicular to transverse axis . It is bisected at the focus . There are two latera recta as there are two focii.


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