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Mathematics. Session. Hyperbola Session - 1. Introduction. If S is the focus, ZZ´ is the directrix and P is any point on the hyperbola, then by definition. Question. Illustrative Problem.

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Session l.jpg
Session

Hyperbola

Session - 1


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Introduction

If S is the focus, ZZ´ is the directrix and P is any point on the hyperbola, then by definition



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Illustrative Problem

Find the equation of hyperbola whose focus is (1, 1), directrix is 3x + 4y + 8 = 0 and eccentricity is 2.

Solution :

Let S(1, 2) be the focus and P(x, y) be any

point on the hyperbola.

where PM = perpendicular distance from P to directrix 3x + 4y + 8 = 0




Definition of special points lines of the equation of hyperbola l.jpg
Definition of Special Points Lines of the Equation of Hyperbola

(ii) Transverse and Conjugate Axes

(i) Vertices

(iii) Foci : As we have discussed earlier S(ae, 0) and S´(–ae, 0) are the foci of the hyperbola.


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(iv) Directrices Hyperbola :The lines zk and z´k´

are two directrices of the hyperbola and

their equations are

respectively.

(vi) Eccentricity

For the hyperbola we have

Definition of Special Points Lines of the Equation of Hyperbola

(v) Centre :The middle point O of AA´ bisects every chord of the hyperbola passing through it and is called the centre of the hyperbola.


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(ix) Focal Distance of a Point Hyperbola

SP = ex – a

S´P = ex + a

Definition of Special Points Lines of the Equation of Hyperbola

(vii) Ordinate and Double ordinate

(viii) Latus rectum

“A hyperbola is the locus of a point which moves in such a way that the difference of its distances from two fixed points (foci) is always constant.”


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The eccentricity of the conjugate hyperbola is Hyperbola

Conjugate Hyperbola

The conjugate hyperbola

of the hyperbola


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Important Terms Hyperbola


Auxiliary circle and eccentric angle parametric coordinate of hyperbola l.jpg

If Hyperbola is the equation

Of hyperbola, then its auxiliary circle is x2 + y2 = a2

= eccentric angle

are known as parametric equation

of hyperbola.

Auxiliary Circle and Eccentric AngleParametric Coordinate of Hyperbola

The circle drawn on transverse axis of the hyperbola as diameter is called an auxiliary circle of the hyperbola.


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inside Hyperbola

Outside

inside

Position of Point with respect to Hyperbola


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Let the equation of line is y = mx + c Hyperbola

and equation of hyperbola is

Intersection of a Line and a Hyperbola

Point of intersection of line and hyperbola could be found out by solving the above two equations simultaneously.


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Intersection of a Line and a Hyperbola Hyperbola

[Putting the value of y in the equation of Hyperbola]

This is a quadratic equation in x and therefore gives two values of x which may be real and distinct, coincident or imaginary.


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Given hyperbola is Hyperbola

and given line is y = mx + c

Condition for Tangency and Equation of Tangent in Slope Form and Point of contact

This is the required condition for tangency.


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Equation of Tangent in Slope Form Hyperbola

Substituting the value of c in the equation y = mx + c, we get equation of tangent in slope form.

Equation of tangent

Point of Contact


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Equation of Tangent and Normal in Point Form Hyperbola

Equation of tangent at any point

(x1, y1)of the hyperbola is

Equation of Normal at any point

(x1, y1)of the hyperbola is


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Equation of tangent at Hyperbola

is

Equation of Tangent and Normal in Parametric Form

Equation of normal in parametric form is


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Class Test Hyperbola


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Class Exercise - 1 Hyperbola

Find the equation to the hyperbola for which eccentricity is 2, one of the focus is (2, 2) and corresponding directrix is x + y – 9 = 0.


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According to the definition of hyperbola Hyperbola

Solution

Let P(x, y) be any point of hyperbola.

Let S(2, 2) be the focus.

This is the required equation of hyperbola.


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Find the coordinates of centre, lengths of the axes, eccentricity, length of latus rectum, coordinates of foci, vertices and equation of directrices of the hyperbola

Class Exercise - 2


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The given equation can be written as eccentricity, length of

Solution


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The equation (i) becomes eccentricity, length of

The coordinates of centre with respect to oldaxes are x – 1 = 0 and y – 2 = 0.

Solution contd..

Shifting the origin at (1, 2) withoutrotating the coordinate axes, i.e.

Put x – 1 = X and y – 2 = Y

Centre: The coordinates of centre with respect to new axes are X = 0 and Y = 0.


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x = 1, y = 2 eccentricity, length of

Solution contd..

Length of axes

Length of transverse axes = 2b

Length of conjugate axes = 2a

Eccentricity


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Foci: eccentricity, length of Coordinates of foci with respect to new axes are, i.e..

Coordinates of foci with respect to old axes are(1, 5) and (1, –1).

Vertices: The coordinates of vertices with respect tonew axes are X = 0 and , i.e. X = 0 and

Solution contd..

Length of latus rectum


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The coordinates of axes with respect to eccentricity, length of old axes are x – 1 = 0, i.e. x = 1 and

Vertices

Directrices: The equation of directrices with respectto new axes are , i.e. .

The equation of directrices with respect toold axesare , i.e. y = 3 and y = 1.

Solution contd..


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Class Exercise - 3 eccentricity, length of

  • Find the equation of hyperbola whose

  • direction of axes are parallel to

  • coordinate axes if

  • vertices are (–8, –1) and (16, –1) and focus is (17, –1) and

  • focus is at (5, 12), vertex at (4, 2) and centre at (3, 2).


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(i) eccentricity, length of Centre of hyperbola is mid-point ofvertices

Equation of hyperbola is

Equation of hyperbola in new coordinate axes is.

Solution

Let x – 4 = X, y + 1 = Y.


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Solution contd.. eccentricity, length of

As per definition of hyperbola

a = Distance between centre and vertices

= 144

Abscissae of focus in new coordinates system isX = ae, i.e. x – 4 = 12e


Solution contd33 l.jpg

Equation of hyperbola is eccentricity, length of

(ii) Coordinates of centre are (3, 2).

Equation of hyperbola is

a = Distance between vertex and centre

Equation (i) becomes

Solution contd..

Let x – 3 = X, y – 2 = Y.


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Abscissae of focus is X = ae eccentricity, length of

5 = e + 3 [Abscissae of focus = 5]

= 1 (4 – 1) = 3

Solution contd..

i.e. x – 3 = e (As a = 1)

x = e + 3


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Class Exercise - 4 eccentricity, length of

Find the equations of the tangents to the hyperbola 4x2 – 9y2 = 36 which are parallel to the line 5x – 3y = 2.


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Tangent is parallel to the given line 5x – 3y = 2 eccentricity, length of

Equation of tangents

Solution


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Find the locus of mid-point of portion of tangent intercepted between the axes for hyperbola

Class Exercise - 5


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Any tangent to the hyperbola intercepted between the axes is

be the middle point of AB

Solution

Let the tangent (i) intersect the x-axis at A and y-axis at B respectively.

Let P(h, k) be the middle point of AB.


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Solution contd.. intercepted between the axes


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Find the condition that the line intercepted between the axes lx + my + n = 0 will be normal to the hyperbola

Class Exercise - 6


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Equations (i) and (ii) will represent the same line if intercepted between the axes

Solution

The equation of the given line is

lx + my + n = 0 ...(ii)


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Solution contd.. intercepted between the axes


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  • The curve intercepted between the axes represents

  • a hyperbola if k < 8

  • an ellipse if k > 8

  • a hyperbola if 8 < k < 12

  • None of these

Class Exercise - 7


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The given equation intercepted between the axes

represents hyperbola if

(12 – k) (8 – k) < 0

i.e. 8 < k < 12

Solution

Hence, answer is (c).


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If the line intercepted between the axes touches the hyperbola at the point , show that

Class Exercise - 8


Solution46 l.jpg
Solution intercepted between the axes


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Both (i) and (ii) represent same line intercepted between the axes

Solution contd..


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Let intercepted between the axes where be two points on thehyperbola If (h, k) is the point of intersection of the normals at P and Q, then k is equal to

(a) (b)

(c) (d)

Class Exercise - 9


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Solution intercepted between the axes


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From (iv) and (v) eliminating h, we get intercepted between the axes

Solution contd..


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Determine the equations of common tangent to the hyperbola intercepted between the axes

Class Exercise - 10


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The equation of other hyperbola can be written as intercepted between the axes

If equations (i) and (ii) are the same, then

Solution


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Solution contd.. intercepted between the axes


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Thank you intercepted between the axes