### Mathematics Class XI

Unit-I: Sets and Functions
Chapter 1: Sets
Unit-II: Algebra
Chapter 5: Binomial Theorem
Unit-III: Coordinate Geometry
Chapter 1: Straight Lines
Chapter 2: Conic Sections
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability
Chapter 1: Statistics
Chapter 2: Probability

# Elementary Concepts of Hyperbola

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Definition:

A hyperbola is defined as the locus of a point moving in a plane in such a way that the ratio of its distance from a fixed point to that from a fixed line is a fixed constant greater than . i.e ( Eccentricity )

Standard equation of hyperbola is , where

Eccentricity

Focii   and

Equation of directrices : and

Transverse Axis:

The line segment of length  in which  the focii and both lie is called the transverse axis of hyperbola.

Conjugate Axis:

The line segment of  length between the two points and is called as conjugate axis of the hypoerbola.

Principal Axes:

The transverse and conjugate axis together are called principal axes of the hyperbola.

Vertices: and

Focal Chord:

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

Double Ordinate:

A chord perpendicular to the transverse axis is called double ordinate.

Latus Rectum (L.R):

The focal chord perpendicular to the transverse axis is called the latus rectum. Its length is given by

Centre:

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

Note:

(i) Length of L.R

(ii) End point of L.R are

Conjugate Hyperbola:

Two hyperbolas are such that the transverse and conjugate axes of the hyperbola are respectively the conjugate and the transverse axes of the other hyperbolas are called conjugate hyperbolas of each other. Example: and are conjugate hyperbolas.

Equation

Vertices:

Note:

(i) If and are the eccentricities of the hyperbola and its conjugate then

(ii) The foci of a hyperbola and its conjugate are concyclic and form the vertices of a square.

(iii) Two hyperbolas are said to be similar if they have the same eccentricity.

(iv) Two similar hyperbolas are said to be equal if they have the same latus rectum.

(v) If a hyperbola is equilateral then the conjugate hyperbola is also equilateral.

Since the fundamental equation to hyperbola only differs from that to ellipse in having instead of , it will be found that many propositions for hyperbola are derived from those for ellipse by simply changing the sign of .

Example:

Find the eccentricity of the hyperbola whose L.R is half of its transverse axis.

Solution:

Equation of hyperbola:

Hence transverse axis and L.R

According to question

Parametric Representation of Hyperbola:

The equations and together represent the hyperbola , where is a parameter.

If is on the hyperbola then is on the auxiliary circle.

The equation to the chord of the hyperbola joining the two-point and is given by

Auxiliary Circle of Hyperbola:

A circle drawn with centre and transverse axis diameter is called the auxiliary circle of the hyperbola. Equation of the auxiliary circle is

Position of the Point w.r.to a Hyperbola:

The quantity is positive, zero or negative according as the point   lies inside, on or outside the curve.

Example:

Find the position of the point relative to hyperbola .

Solution:

So the point lies inside the hyperbola

Rectangular Hyperbola (Equilateral Hyperbola)

The particular kind of hyperbola in which the lengths of the transverse and conjugate axis are equally called a rectangular hyperbola/ equilateral hyperbola.

Since equation of hyperbola is , whose asymptotes are Eccentricity

Rotation of this system through an angle of in clockwise direction gives another form of the equation of rectangular hyperbola which , where .

it is referred to as its asymptotes as axes of co-ordinates.

Vertices: and

Focii:   and

Directrices:

Latus Rectum: = Transversal Axis = Conjugate Axis

Parametric Equation :

Line and a Hyperbola:

The straight line is a secant, a tangent or passes outside the hyperbola according as or or

Note:

The equation of the chord of the hyperbola   joining the two points and given by

Line and a Rectangular Hyperbola:

Equation of a chord joining the points and is .

Equation of the tangent at is and at is

Tangent to Hyperbola:

(i) Slope form:

can be taken as the tangent to the hyperbola , having slope .

(ii) Point Form:

Equation of tangent to the hyperbola at the point is

(iii) Parametric Form:

Equation of the tangent to the hyperbola at the point is

Note:

(i) Point of intersection o the tangents at and   is

(ii) If , then tangents at these points are parallel.

(iii) There are two parallel tangents having the same slope . These tangents touch the hyperbola at extremities of a diameter.

Example:

Find , if touch the hyperbola

Solution:

Normal to Hyperbola:

(i) The equation of the normal to the hyperbola   at the point on it is

(ii) The equation of the normal at the point on the hyperbola is

(iii) Equation of normals in term of their slopes are

(iv) Normal to a rectangular hyperbola

Equation of the normal at is

Some important properties of Hyperbola:

(i) Difference of focal distances is a constant. i.e

(ii) Locus of the feet of the perpendicular drawn from focus of the hyperbola upon any tangents is its auxiliary circle, i.e.,   and the product of these perpendicular is .

(iii) The portion of the tangent between the point of contact and the directrix subtends a right angle at the corresponding focus.

(iv) The focii of the hyperbola and the points and in which any tangent at the vertices are concyclic with as diameter of the circle.

Comparative Study of Two Hyperbolas:

 Hyperbola 1 Centre 2 Foci 3 Directices 4 Vertices 5 Axes Transverse axis, Conjugate Axis, Transverse Axis, Conjugate Axis , 6 Latus Rectum 7 Relation between and
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