ram rajya more, siwan xi th , xii th , target iit-jee (main + … · 2019-11-13 · definition of...

FIROZ AHMADʼS MATHEMATICS

Ram Rajya More, Siwan (Bihar)M.Sc. (Maths), B.Ed, M.Phil (Maths)

MATHEMATICSMob. : 9470844028

9546359990

M.Sc. (Maths), B.Ed, M.Phil (Maths)

RAM RAJYA MORE, SIWAN

XIth, XIIth, TARGET IIT-JEE(MAIN + ADVANCE) & COMPATETIVE EXAM

FOR XI (PQRS)

1

Key Concept - I ....................................................................................

Exericies-I .....................................................................................

Exericies-II .....................................................................................

Exericies-III .....................................................................................

Solution Exercise

Page .....................................................................................

CONTENTS

HYPERBOLA& Their Properties



THINGS TO REMEMBER Definition

A hyperbola is the set of point in a plane whose distance from two fixed points in the plane have aconsatnt difference. The two fixed points are the foci of the hyperboal.

OR

A hyperbola is the locus of a point in a plane which moves in a plane in such a way that the ratio of itsdistance from a fixed point (ie, focus) to the distance from a fixed line (ie, directrix) is greater then unity.

Mathematically, PMSP

= e, where e > 1.

Definition of Terms Related to an Ellipes

Vertices

The point A and A’, where the curve meets the line joining the foci S and S’, are called the vertices of thehyperbola.

Transverse and Conjugate Axes

Transverse axis is the one which the curve meets the line passing thorugh the foci and perpendicular tothe directrices and conjugate axis is the one which is perpendicular to the transverse axis and passes throughthe mid point of the foci (ie, centre).

Centre

The mid point C of AA’ bisects every chord of the hyperbola passing throgh it and is called the centre ofthe hyperbola.

Double Ordinate

If Q be a point on the hyperbola draw QN perpendicular to the axis of the hyperbola and produced tomeet the curve again at Q’. Then, QQ’ is called a double ordinete of Q.

Latusrectum

The double ordinate passing through focus is called latusrectum.

2



3

Terms Related to an Hyperboal

(a) Equation 12

2

2

2

by

ax 12

2

2

2

by

ax

(b) Graph

(c) Centre C (0, 0) C (0, 0)

(d) Vertices (+a, 0) (0, +b)

(e) Length of transverse axis 2a 2b

(f) Length of conjugate axis 2b 2a

(g) Foci (+ae, 0) (0, +be)

(h) Equation of direction x =

ea

y =

eb

(i) Eccentricity b2 =a2(e2 – 1) b2 =a2(e2 – 1)

(j) Length of latusrectumab22

ba22

(k) Ends of latusrectum

abae

2

,

be

ba ,

2

(l) Parametric equations 22

tansec

eeyandeeaxor

byax

sectan

byax

Type of Conic Ellipse Conjugate Ellipse



4

Position of a Point with Respect to an Hyperboal

Let the equatio of a hyperboal is 12

2

2

2

by

ax

and coordinat of a point P are (h, k). Then

Point P lies outside, if

012

2

2

2

bk

ah

Point P lies on the hyperbola, if, if

012

2

2

2

bk

ah

and point P lies inside if

012

2

2

2

bk

ah

Intersection of aline a Hyperbola

Let the hyperbola be 12

2

2

2

by

ax ...(i)

and the given line be y = mx + c ....(ii)

On eliminating y from Eqs. (i) and (ii)

1)(2

2

2

2

b

cmxax

(m) Parametric coordinates (a sec , b tan ) (a tan , b sec )

(n) Focal radii | SP | = (ex1 – a ) and | S’P | = (ex1 + a) | SP | = (ey1 – a ) and | S’P | = (ey1 + a)

(o) Sum of focal radii | SP | + | S’P | 2a 2b

(p) Distance betwen foci 2ae 2be

(q) Tangents at vertices x = a and x = –a x = b and y = –b



b2x2 – a2(mx + c)2 = a2b2

(a2m2 – b2)x2 + 2mca2x + a2(b2 + c2) = 0 ...(iii)

If is a quadratic equation in x which gives two values of x.

D = 4m2c2a4 – 4(a2m2 – b2)a2(b2 + c2)

= –a2m2 + b2 + c2

Now, If D > 0 ie, c2 > a2m2 – b2, then line intersect the hyperbola at two points.

If D = 0 ie, c2 = a2m2 – b2, then line intersect the hyperbola at one points.

And if D < 0 ie, c2 < a2m2 – b2, then line neither touch nor intersect the hyperbola.

Condition of Tangency

The line y = mx + c will be touch a hyperbola 12

2

2

2

by

ax

,if

c2 = a2m2 + b2

and coordinates of point of contact are

222

2

222

2

,bma

bbma

ma.

Equation of Tangent in Different Forms

1. Point Form

The equation of tangent to the hyperbola 12

2

2

2

by

ax at the point (x1, y1) is 12

121

byy

axx .

2. Slop Form


2

2

2

by

ax in the slope form is y = mx + 222 bma .

3. Parametric Form


2

2

2

by

ax at a point(a sec , b sin ) is

ax sec +

ay

tan = 1.

Equation of Pair to Tangents

Let P(x1, y1) be a point which lies outside the hyperbola 12

2

2

2

by

ax , then the eqution of pair of

tangens drawn from external point P to the hyperbola is given by

SS1 = T2

Where, S = 12

2

2

2

by

ax , S1 = 12

21

2

21

by

ax

and T = 121

21

byy

axx

5



Chord of Contact

Let PA and PB be any two tangents to the hyerbola from a point P(x1, y1), then AB is known as chordof contact and its equation is given by

T = 0

121

21

byy

axx

= 0

Equation of the chord Biscted at a Given Point.

The equation of the chord of the hyperbola 12

2

2

2

by

ax bisected at the pont (x1, y1) is

121

21

byy

axx

= 12

21

2

21

by

ax

Director Circle

The locus of the point of intersection of the tangents to the hyperbola 12

2

2

2

by

ax , which are

perpendicular to each other is called directro circle. The equation of the director circle of the hyperbola

12

2

2

2

by

ax is x2 + y2 = a2 – b2.

Equation of Normal in Different Forms

1. Point Form

The equation of normal at the point (x1, y1) to the hyperbola 12

2

2

2

by

ax is 12

121

byy

axx .

2. Slop Form

The equation of normal of slope m to the hyperbola 12

2

2

2

by

ax is y =

222

22 )(mba

bammx

at the

points

222

2

222

2

,mba

mbmba

a .

6



7

3. Parametric Form

The equation of normal to the hyperbola 12

2

2

2

by

ax at (a sec , b tan ) is ax cos – by cot =

a2 + b2.

Number of Normals and Conormal Point

There are exatly four lines passing through a given point such that they are normals ot the hyperbolaat eh points where they intersect the hyperbola. Such points on the hyperbola are known as the conormalpoints.

Properties of E ccentric Angle of Conormal Points

(a) The sum of the eccentric angles of conormal points is an odd multiple of p.

(b) If 1, 2, 3 and 4 be eccentric angle of four points on the hyperboal 12

2

2

2

by

ax , the normals at

which are concurrent, then

(i) cos (1 + 2) = 0.

(ii) sin (1 + 2) = 0.

Asymptotes

An asymptote of any hyperbola is a straight line which touches in it two points at infinity. In otherwords asympotes are the lines which are tangents to the curve at infinity.

The equation of two asymptotes of the hyperbola 12

2

2

2

by

ax are y = + x

ab

.

Rectangular Hyperbola

A hyperbola whose asymptotes are at right angles to each other, is said to be rectangular hyperbola.

If the length of transverse and conjugate axes of any hyperbola are equal, then hyperbola is known asrectangular hyperbola.

The equation of rectangular hyperbola to the hyperbola 12

2

2

2

by

ax is x2 – y2 = a2 and equation of the

asymptotes are y = + x is y = x and y = –x. Clearly, each of these two asymptotes is inclined at 45o to thetransverse axis.



8

When the centre of any rectangular hyperbola be at the origin and its asymptotes conicide with the

coordinate axes, then the equation of rectangular hyperbola is xy = c2. Its eccentricity is 2 .

Properties of Rectangular Hyerbola xy = c2

(a) Equation of the chord joining ‘t1’ and ‘t2’ is x + yt1t2 – c(t1 + t2) = 0.

(b) Equation of tangent at, (x1, y1) is xy1 + x1y = 2c2.

(c) Equation of tangent at

tcct, is

tx + yt = 2c.

(d) Point of intersection of tangents at t1 and t2 is

2121

21 2,2tt

ctttct

.

(e) Equation of normal at

tcct, is xt3 – yt – ct4 + c = 0.

(f) Equation of normal to rectangular hyperbola at (x1, y1) is

xx1 – yy1 = 21

21 yx .



Note :

The vertex divides the join of focus and the point of intersection of directrix with axis internally andexternally in the ratio e : 1.

Domain and rang of a hyperbola 12

2

2

2

by

ax are x < –a or x > a and y R respectively..

The tangents at the points P (a sec, b tan 1) and (a sec 2, b tan 2) intersect at the point R

2cos

2sin

,

2cos

2cos

21

21

21

21

ba

For director circle of 12

2

2

2

by

ax a must be greater then b. If a < b, then director circle x2 + y2 = a2 – b2

does not exist.

The line y = mx + c will be a normal to the hyperbola 12

2

2

2

by

ax , if 222

2222 )(

mbabamc

.

For normals can be drawn to the hyperbola from any point outside to .... hyperbola.

A hyperbola and its conjugate hyperbola have the same asymptotes.

The angle between the asymptotes of 12

2

2

2

by

ax is 2 tan–1

ab

.

The asymptotes pass through the centre of the hyperbola

The bisectors of the angle between the asymptotes are the coordinate axis.

The product of the perpendicular from any point on the hyperbola 12

2

2

2

by

ax to its asymptotes is

equal to 22

22

baba

9

ram rajya more, siwan xi th , xii th , target iit-jee (main + … · 2019-11-13 · definition of...

Documents