hyperbola jee mains notes

Standard Equation of Hyperbola

The equation of the hyperbola is simplest when the centre of the hyperbola is at the origin, and the foci are either on the x-axis or on the y-axis.  The standard equation of a hyperbola is given as follows:

[(x² / a²) – (y² / b²)] = 1

where , b² = a² (e²  – 1)

Important Terms and Formulas of Hyperbola

There are certain terms related to a hyperbola which need to be thoroughly understood to be able to get confident with this concept. Some of the most important terms related to hyperbola are:

• Eccentricity (e): e² = 1 + (b² / a²) = 1 + [(conjugate axis)² / (transverse axis)²]
• Focii: S = (ae, 0) & S′ = (−ae, 0)
• Directrix: x=(a/e), x = (−a / e)
• Transverse axis:

The live segment A’A of length 2a in which the foci S’ and S both lie is called the transverse axis of the hyperbola.

• Conjugate axis:

The line segment B’B of length 2b between the 2 points B’ = (0, -b) & B = (0, b) is called the conjugate axis of the hyperbola.

• Principal axes:

The transverse axis and conjugate axis.

• Vertices:

A = (a, 0) & A’ = (-a, 0)

• Focal chord:

A chord which passes through a focus is called a focal chord.

• Double ordinate:

The chord perpendicular to the transverse axis is called a double ordinate.

• Latus rectum:

The focal chord ⊥^r to the transverse axis is called the latus rectum.

Its length = (2b² / a) = [(conjugate)² / transverse] = 2a (e² − 1)

The difference in focal distances is a constant

i.e., |PS−PS′| = 2a

Length of latus rectum = 2 e × (distance of focus from the corresponding directrix)

End points of L.R: (± ae, ± b² / a)

Centre:

The point which bisects every chord of the conic, drawn through it, is called the centre of the conic.

C: (0, 0) is the centre of [(x² / a²) – (y² / b²)] = 1