Relation between Focal Distances

 

The difference of the focal distances of a point on the hyperbola is constant. PM and PM’ are perpendiculars to the directrices MZ and M’Z’ and PS’ – PS = e(PM’ – PM) = eMM’ = e(2a/e) = 2a = constant.

Another Definition of Hyperbola

 A hyperbola can be defined in another way; Locus of a moving point such that the difference of its distances from two fixed points is constant, would be a hyperbola.

Transverse and Conjugate axes:

The points A(a, 0) & A’(–a, 0) are called the vertices of the hyperbola and the line AA’ joining the vertices is called the transverse axis and the line perpendicular to it, through the centre (0, 0) of he hyperbola is called conjugate axis.

Centre:

Any chord of the hyperbola through C will be bisected at C (by symmetry), therefore C is called the centre of hyperbola.

Thus hyperbola is a central curve.

Latus rectum:

The chord of a hyperbola through one of the foci and at right angle to the transverse axis is called the latus rectum:

∴ If 2l be the length of the latus rectum, then the co-ordinates of one of its extremities is (ae, l).

∴ The point (ae, l) lies on the hyperbola x²/a² – y²/b², so we have

        e² (l²/b²) = 1

⇒ l² = b²(e²–1) = b²a²(e²–1)/a² = b⁴/a² (? b² = a²(e² – 1)

⇒ l = b²/a²

∴ The length of the latus rectum = 2 b²/a²

Relative Position of a Point with respect to the Hyperbola

The quantity x₁²/a² – y₁²/b² = 1 is positive, zero or negative, according as the point (x₁, y₁) lies within, upon or without the curve.

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