A hyperbola is the locus of a point which moves such that, ratio of its distance from a fixed point (focus) and its distance from a fixed straight line (directrix), is a constant (eccentricity).
This constant (eccentricity) is greater than unity.
Standard Equation and Basic Definitions:
Let S be the focus and ZM the directrix of a hyperbola.
Since e > 1, we can divide SZ internally and externally in the ratio e : 1, let the points of division be A and A’ as in the figure.
Let AA’ = 2a and C is the mid point of AA’
Then, SA = e. AZ
and SA’ = e. ZA’
⇒ SA + SA’ = e(AZ + ZA’)
i.e., 2SC = 2ae or SC= ae.
Similarly by subtraction,
SA’ − SA = e(ZA’ − ZA) = 2eZC
⇒ 2a = 2eZC
⇒ ZC = a/e.
Now, take C as the origin , CS as the x-axis, and the perpendicular line CY as the y axis.
Then, S is the point (ae , 0) and ZM the line x = a/e.
Let P(x , y) be any point on the hyperbola.
Then the condition PS2 = e2. (distance of P from ZM)2 gives
(x − ae)2 + y2 = e2 (x − a/e)2
or x2(1 – e)2 + y2 = a2(1 – e2)
Since e > 1, e2 – 1 is positive.
Let a2(e2 − 1) = b2 .
Then the equation (i) becomes
∎ The eccentricity e of the hyperbola is given by the relation
∎ Since the curve is symmetrical about the y – axis, it is clear that there exists another focus S’ at (−ae, 0) and a corresponding directirx Z’M’ with the equation x= −a/e , such that the same hyperbola is described if a point moves so that its distance from S’ is e times its distance from Z’M’.
∎ The points A and A’ where the straight line joining the two foci cuts the hyperbola are called the vertices of the hyperbola.
∎ The straight line joining the vertices is called the transverse axis of the hyperbola, its length AA’ is 2a
∎ The middle point C of AA’ possesses the property that it bisects every chord of the hyperbola passing through it. It can be proved by taking P(x1 , y1) as any point on hyperbola.
If (x1 , y1) lies on the hyperbola then so does P'(−x1 , −y1) because hyperbola is symmetric about x and y axes. Therefore PP’ is a chord whose middle point is (0, 0), i.e. the origin O. On account of this property the middle point of the straight line joining the vertices of the hyperbola is called the centre of the hyperbola.
∎ The straight line through the centre of a hyperbola which is perpendicular to the transverse axis does not meet the hyperbola in real points. If B and B’ be the points on this line such that BC = CB’ = b , the line BB’ is called the conjugate axis.
∎ The latus rectum is the chord through a focus at right angle to the transverse axis.
∎ The length of the semi-latus rectum obtained by putting
x = ae in the equation of the hyperbola is