Let S ≡ = 0 be a standard form of the hyperbola.

where a > 0, b > 0 and b^{2} = a^{2} (e^{2} – 1)

i) If y = 0, then x = ± a

It means the hyperbola cuts the X-axis at two points A (a, 0) and A' (– a, 0)

ii) If x = 0, then y = ± √ (– b^{2})

Since y is imaginary, the point does not exist in the Cartesian Plane.

Therefore, the curve does not intersect the Y-axis.

iii) Rewriting the equation S = 0 as y = ±

we can see that y is real.

∴ x^{2} – a^{2} ≥ 0 ⇒ x ≤ – a or x ≥ a

It means the curve does not exist between the two vertical lines x = a and x = – a.

Further from x = ± we can conclude that x is real for all values of y.

Hence each
horizontal line y = k intersects the hyperbola at two points.

Further, as y → ± ∞, x also tends to ± ∞. i.e, the curve has no bounds (or limits).

iv) For any value x ∈ **R**, we have two values of y – equal in magnitude but opposite in sign.

(because y = ± ). Therefore, the curve is symmetric about X-axis.

v) For each real value of y, we have two values of x – again equal in magnitude but opposite in sign. (because x = ± )

Therefore, the curve is also symmetric about Y-axis.

We therefore, conclude that the curve consist of two symmetrical branches – each extending to infinity in two directions.

vi) along X-axis is called the **transverse axis ** of the hyperbola and (along Y-axis) is its **conjugate axis**. Refer above figure.

The curve does not meet conjugate axis.

vii) The curve has two foci and two directrices. (This was also the case with an ellipse).

Foci : S(ae, 0) and S'(– ae, 0)

Equations of directrices : x = ± a/e

viii) The point of intersection of the transverse and conjugate axes is the **centre** of the hyperbola. The centre (C) bisects every chord of the hyperbola that passes through itself.

Let S ≡ = 0 be a hyperbola.

Then S' ≡ = 0 will be its conjugate hyperbola.

Their properties are described below under two cases:

i) centre at the origin

ii) centre not at the origin

## Centre at the origin

Hyperbola |
Conjugate hyperbola |

1. Transverse axis is along X-axis (y = 0), Length of the transverse axis is 2a |
Transverse axis is along Y-axis (x = 0), Length of the transverse axis is 2b |

2. Conjugate axis is along y-axis (x = 0), Length of conjugate axis is 2b |
Conjugate axis is along X-axis (y = 0), Length of conjugate axis is 2a |

3. Coordinates of the centre (0, 0) | Coordinates of the centre (0, 0) |

4. Coordinates of the foci (± ae, 0) | Coordinates of the foci (0, ± be) |

5. Equation of the directrices x = ± a/e | Equation of the directrices x = ± b/e |

6. Eccentricity e = | Eccentricity e' = |

Hyperbola |
Conjugate hyperbola |

1. Transverse axis is along y = k, Length of the transverse axis is 2a |
Transverse axis is along x = h, Length of the transverse axis is 2b |

2. Conjugate axis is along x = h, Length of conjugate axis is 2b |
Conjugate axis is along y = k, Length of conjugate axis is 2a |

3. Coordinates of the centre (h, k) | Coordinates of the centre (h, k) |

4. Coordinates of the foci (h ± ae, k) | Coordinates of the foci (h, k ± be) |

5. Equation of the directrices x = h ± a/e | Equation of the directrices x = k ± b/e |

6. Eccentricity e = | Eccentricity e' = |

**Note: ** For a **Rectangular hyperbola** x^{2} – y^{2} = a^{2} (or) y^{2} – x^{2} = a^{2} – the foci, centre etc are obtained by substituting b = a and e = √2 (in the above tables).

**Chord:** A line segment joining the points on the hyperbola is called a 'chord'.

**Double ordinate:** A chord passing through a point P on the hyperbola and perpendicular to the transverse axis (principal axis) of the hyperbola is called the double ordinate.

**Focal chord:** A chord passes through one of the foci is called a focal chord.

**Latus rectum:** A focal chord of a hyperbola perpendicular to the transverse axis (principal axis) of the hyperbola is called latus rectum.

If the latus rectum meets the hyperbola in L and L' then LL' is called the length of the latus rectum.

Let L and L' be the end points of the latus rectum of the hyperbola .

Let SL = *l*, then L = (ae, *l*)

Since L lies on the hyperbola | ⇒ | |

⇒ | ||

⇒ | ||

⇒ | ||

⇒ | ||

∴ The length of the latus rectum (LL') | = | 2(SL) |

= |

**Note: ** The end points of the latera recta are .

If the lengths of the transverse axis (2a) and the conjugate axis (2b) are equal, the hyperbola is specially called a **rectangular hyperbola** (i.e., when a = b).

So the equation of rectangular hyperbola is ** x ^{2} – y^{2} = a^{2} **

And its eccentricity, e = √2

## Auxiliary circle of hyperbola

The circle described on the transverse axis of a hyperbola as diameter is called its **auxiliary circle**. Its equation is **x ^{2} + y^{2} = a^{2}**.

## Parametric equations of hyperbola

Let 'θ' be a parameter such that 0 ≤ θ ≤ 2π; θ ≠ ; θ ≠ ;

Let P(x, y) be any point on the hyperbola . Then **x = a sec θ** and **y = b tan θ** are called the parametric equations of a hyperbola. The point P(a sec θ, b tan θ) is denoted by P(θ) or simply θ.

**Note: ** Let 'θ' be a parameter and P(x, y) be any point on the hyperbola can also be represented as (a cos hθ, b sin hθ). The equations **x = a cos hθ, y = b sin hθ** are also called Parametric equations of hyperbola.

The hyperbola divides the cartesian plane in two disjoint parts. The part containing the foci is called the interior (or inside) of the hyperbola. The remaining part is called the exterior (or inside) of the hyperbola.

Let P (x_{1}, y_{1}) be a point and be a hyperbola then

i) S_{11} > 0 ⇔ P lies inside the hyperbola

ii) S_{11} = 0 ⇔ P lies on the hyperbola

iii) S_{11} < 0 ⇔ P lies outside the hyperbola

Thus the point P lies inside, on or outside the hyperbola S = 0 according as S_{11} is positive, zero (or)
negative i.e., S_{11} ⋛ 0.

## Tangent:

Let S = 0 be a hyperbola and P be a point on the hyperbola. Let Q be any other point on the hyperbola. If the secant line approaches to the same limiting position as Q moves along the curve and approaches to P from either side, then the limiting position is called a **tangent line** or **tangent** to the hyperbola at P. The point P is called point of contact of the tangent to the hyperbola. The equation of tangent at P w.r.t. the hyperbola S = 0 is S_{1} = 0. Refer Fig (i).

## Normal:

Let S = 0 be a hyperbola and P be a point on the hyperbola S = 0. The line passing through P and perpendicular to the tangent of S = 0 at P is called the **normal** to the hyperbola S = 0 at P. Refer Fig (ii).

Let P (x_{1}, y_{1}) be an external point to the hyperbola S ≡ = 0, then pair of tangents PQ, PR can be drawn to it from P. The equation of pair of tangents PQ and PR is

**S _{1}^{2} = S.S_{11}**.

## Director circle

The point of intersection of perpendicular tangents to a hyperbola lies on a circle, concentric with the hyperbola is called the **director circle** of the hyperbola. The equation of the director circle of the hyperbola is **x ^{2} + y^{2} = a^{2} – b^{2}** and its radius is .

Let .... (i) be a hyperbola

and y = mx + c .... (ii) be a straight line

Substituting (ii) in (i), we get

⇒ x^{2} (a^{2} m^{2} – b^{2}) + 2a^{2}mcx + a^{2} (c^{2} + b^{2}) = 0 .... (iii)

Let x_{1} and x_{2} be the roots of the quadratic equation (iii) and D its discriminant.

D | = | 4 a^{4} m^{2} c^{2} – 4 (a^{2} m^{2} – b^{2}) (a^{2} c^{2} + a^{2} b^{2}) |

D | = | 4 a^{2} c^{2} b^{2} – 4 a^{4} b^{2} m^{2} + 4 b^{4} a^{2} |

D | = | 4 a^{2} b^{2} (c^{2} – a^{2} m^{2} + b^{2}) |

i) If D > 0 i.e, c^{2} > a^{2} m^{2} – b^{2}, then the roots are real and distinct.

In this case (ii) is not a tangent to (i). Refer Fig. (i)

ii) If D = 0 i.e, c^{2} = a^{2} m^{2} – b^{2}, then the roots are coincidence (equal).

In this case (ii) is a tangent to (i). Refer Fig. (ii)

iii) If D < 0 i.e, c^{2} < a^{2} m^{2} – b^{2}, then the roots are imaginary.

In this case (ii) is not a tangent to (i). Refer Fig. (iii)

## Note:

i) The condition for a straight line y = mx + c to be a tangent to the hyperbola is ** c ^{2} = a^{2} m^{2} – b^{2}**.

ii) The equation of a tangent to the hyperbola may be taken as and the point of contact is where c

^{2}= a

^{2}m

^{2}– b

^{2}.

iii) The equation of the tangent at ‘θ’ is (θ ≠ , ).

iv) The equation of the normal at P(x

_{1}, y

_{1}) is

**= a**(y

^{2}+ b^{2}_{1}≠ 0) which is always the case except at vertices. At vertices X-axis is the normal.

v) The equation of the normal at ‘θ’ is

**= a**(θ ≠ 0, π).

^{2}+ b^{2} Let PQ and PR be tangents to the hyperbola S ≡ = 0 drawn from an external point P (x_{1}, y_{1}). Then the equation of chord of contact QR is = 0 or **S _{1} = 0**.

## Equation of the chord of the hyperbola with mid point

Equation of the chord of the hyperbola S ≡ = 0, bisect at the given point (x_{1}, y_{1}) is **S _{1} = S_{11}** i.e., .

**Note: ** The equation of the chord joining the points P (a sec α, b tan α) and Q (a sec β, b tan β) is .

The locus of the point of intersection of the tangents to the hyperbola at A and B is called the **polar** of the point P w.r.t. the hyperbola and the point P is called the **pole** of the polar. The equation of the polar with (x_{1}, y_{1}) as its pole is S_{1} = 0 i.e., = 0.

The pole of a line lx + my + n = 0 w.r.t. the hyperbola is

## Properties of Pole and Polar

i) If the Polar P (x_{1}, y_{1}) passes through Q (x_{2}, y_{2}), then the polar of Q (x_{2}, y_{2}) goes through P (x_{1}, y_{1}) and such points are said to be **conjugate points**.

ii) If P (x_{1}, y_{1}), Q (x_{2}, y_{2}) are two conjugate points w.r.t. the hyperbola S = 0, then **S _{12} = 0**. i.e,

iii) If the pole of a line lx + my + n = 0 lies on the another line l'x + m'y + n' = 0, then the pole of the second line lies on the first and such lines are said to be

**conjugate lines**.

iv) Pole of a given line is same as point of intersection of tangents as its extremities.

v) Polar of locus is directrix.

The locus of the mid point of a system of parallel chords of a hyperbola is called a **diameter** of the hyperbola and the point where the diameter intersects the hyperbola is called the **vertex** of the diameter.

Let y = mx + c be a system of parallel chords to the hyperbola for different chords then the equation of diameter of the hyperbola is , which is passing through (0, 0). Refer Fig (i).

## Conjugate diameter

Two diameters an said to be conjugate when each bisects all chords parallel in the others.

If y = m_{1}x, y = m_{2} x be conjugate diameters then **m _{1}m_{2} = **. Refer Fig (ii).

1. Hyperbola shape is extensively used in the design of bridges.

Ex: Twin Arch 138 at Ichinomiya, Japan

2. Open orbits of some comets about the Sun follow hyperbolas.

3. Interference pattern produced by two circular waves is hyperbolic in nature.

4. It is the basis for solving trilateration problems.

5. Hyperbolas are seen in sundials and in shadow of a lampshade or flashlight.

**To know what is a hyperboloid and a hyperbolic-paraboloid click **