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Hyperbola

Hyperbola

A hyperbola is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point in the same plane to its distance from a fixed line is always constant which is always greater than unity.
Hyperbola 1

Standard equation of the hyperbola

Let S be the focus, ZM be the directrix and e be the eccentricity of the hyperbola, then by definition, Hyperbola 2, where b2 = a2(e2 − 1).
Hyperbola 3

Conjugate hyperbola

The hyperbola whose transverse and conjugate axis are respectively the conjugate and transverse axis of a given hyperbola is called conjugate hyperbola of the given hyperbola.
Hyperbola 4Difference between both hyperbolas will be clear from the following table:
Hyperbola 5

Special form of hyperbola

If the centre of hyperbola is (h, k) and axes are parallel to the co-ordinate axes, then its equation is Hyperbola 6.

Auxiliary circle of hyperbola

Let Hyperbola 7 be the hyperbola, then equation of the auxiliary circle is x2 + y2 = a2.
Let ∠QCN = Ï•. Here P and Q are the corresponding points on the hyperbola and the auxiliary circle (0 ≤ Ï• < 2Ï€).
Hyperbola 8

Parametric equations of hyperbola

The equations x = a sec Ï• and y = b tan Ï• are known as the parametric equations of the hyperbola Hyperbola 9.
This (a sec Ï•, b tan Ï•) lies on the hyperbola for all values of Ï•.

Position of a point with respect to a hyperbola

Hyperbola 10

Intersection of a line and a hyperbola

Hyperbola 11

Equations of tangent in different forms

Hyperbola 12

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