Understanding the equation of hyperbola.

Understand the Equation of Hyperbola. Click on the Link to Watch the VIDEO explanation: 
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Equation of the Hyperbola 
Equation of the hyperbola in the form of x square by a square minus y square by b square is equal to 1 
Hyperbola is the locus of a point which moves such that the ratio of its distance from the focus to its distance from the directrix is greater than 1. 
S is the focus and l is the directrix. Draw SZ perpendicular to l. 
On SZ mark the points A and A dash such that 
SA by AZ is equal to SA dash by A dash Z equal to e by 1 
Therefore, SA is equal to eAZ let this be the equation 1 
SA dash is equal to eA dash Z let this be the equation 2 
Bisects AA dash at C. 
Take C as the origin, CS produced as X - axis and CY perpendicular to CS as Y - axis. 
Let P (x, y) be any point on the hyperbola. Join PS. 
Draw PM is perpendicular to the directrix, PN is perpendicular to x - axis. Take CA equal to CA dash equal to a. 
Adding equations 1 and 2 
SA plus SA dash is equal to e into AZ plus A dash Z 
i.e., CA minus CA plus CS plus CA is equal to e into AA dash 
i.e., 2cs is equal to e into 2a since AA dash is equal to 2a 
Therefore, CS is equal to ae. Therefore, Coordinates of S are ae,0 
Subtracting 1 from equation 2, then SA dash minus SA is equal to e into A dash Z minus AZ 
i.e., AA dash is equal to e into CZ plus CA dash minus CA minus CZ 
i.e., 2a is equal to e. 2cz. Therefore, CZ is equal to a by e. 
Therefore Coordinates of z are a by e , 0 
By distance formula, PS is equal to root x minus ae whole square plus y minus 0 whole square 
From the fig. , PM is equal to NZ is equal to CN minus CZ is equal to x minus a by e Since CN is equal to x 
Therefore, PM is equal to x minus a by e 
Since P is a point on the hyperbola, 
PS by PM is equal to e i.e., Ps is equal to e.PM 
Therefore root of x minus ae the whole square plus y minus 0 the whole square is equal to e into x minus a by e 
i.e., root of x minus ae the whole square plus y square is equal to ex minus a 
Squaring both sides and simplifying 
x square into 1 minus e square plus y square is equal to a square 1 minus e square 
Multiply by minus 1 
minus x square into 1 minus e square minus y square is equal to minus a square into 1 minus e square 
i.e., x square into e square minus 1 minus y square is equal to a square into e square minus 1 
x square by a square minus y square by a square into e square minus 1 is equal to 1 
Since, e is greater 1, a square into e square minus 1 is positive. 
Therefore, Take a square into e square minus 1 is equal to b square 
Therefore, x square by a square minus y square by b square is equal to 1. 
This is the equation of the hyperbola.