ram rajya more, siwan xi th , xii th , target iit-jee (main +...

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

PARABOLA& Their Properties



THINGS TO REMEMBER Conic Section

The locus of a point P shich moves in a plane such that its distance from a fixed point is always in aconstant ratio to its perpendicular distance from a fixed straight line, is known as conic section.

The fixed point is called the focus of the conic and this fixed line is called the directrix of the conic.Also this constant ratio is called the eccentricity of the conic and is denoted by by e, Thus

PMPS = constant = e PS = ePM

Equation of Conic Section

If the focus is (, ) and the directrix is ax + by + c = 0, then the equation of the conic section whoseeccentricity is e, is

22

22

ba

cbyaxeyx

22

2222

bacbyaxeyx

Some Important Definitions

1. Centre

The point which bisects every chord of the conic passing through it, is called the center of theconic section.

2

Eccentricity Shape

e = 0

0 < e < 1

e = 1

e > 1

e =

Circle

Ellipse

Parabola

Hyperbola

Pair of straight lines

M P

S (Focus)Dire

ctrix

M P(x, y)

S (, )

ax +

by

+ x

= 0



2. Axis

The straight line passing through the focus and perpendicular to the directrix, is called axis of theconic section.

3. Vertex

The points of intersection of the coinc section and the axis, is called the vertex of conic section.

4. Latusrectum

The chord passing through the focus and perpenicular to the axis is called latusrectum of conicsection.

5. Focal Chord

A chord of a conic passing through the focus is called a focal chord.

6. Double Ordinate

A straight line drawn perpendicular to the axis and terminated at both ends of the curve is adouble ordinate of the conic section.

Recognisation of Conics

The equation of conics represented by general equation of second degree

ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 ...(i)

can be reconised easily by the condition given below.

Case I When = abc + 2fgh – af2 – bg2 – ch2 = 0

In this case Eq. (i) represents the degenerate conic.

Case II = abc + 2fgh – af2 – bg2 – ch2 0

In this case Eq. (i) represents the non-degenerate conic.

A pair of straight lines or empty set.

A pair of intersecting straight lines.

Real or Imaginary pair of staright lines.

Point.

Nature of ConicCondition

= 0 and ab – h2 = 0

= 0 and ab – h2 0

= 0 and ab – h2 < 0

= 0 and ab – h2 > 0

A circle

A parabola

An ellipse or empty set

A hyperbola

A rectangular hyperbola

Nature of ConicCondition

0, h = 0, a = b

0, ab – h2 0

0, ab – h2 > 0

0, ab – h2 < 0

0, ab – h2 < 0 and a + = 0

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Centre of Conics

Centre of the conic is the point which bisect every chord of the conic passing through it. If the equation

of conic is ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, then the coordinates of centre of conic are

22 ,habafgh

habbghf

.

Parabola

A parabola is the locus of a point which moves in a plane such that its distance from a fixed point (ie,focus) is always equal to its distance from a fixed, straight line (ie, directrix).

Mathematically, 1 ePMPS

Where, e is called eccentricity.

Terms Related to all Parabolas (in Standard Form)

(a)circle :plane

perpenicularto cone axis

(b)Ellipse : plane not

perpenicular toaxis not parallel to

side of cone

(c)Parabola :

planeparallel to

side of cone

(d)Hyperbola :

planeparallel tocone axis

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(a) Equation of parabol

(b) Graph

(c) Ennentricity

(d) Focus

y2 = 4ax y2 = –4ax x2 = 4ay x2 = –4ay

e =1

S(a, 0)

e =1

S(–a, 0)

e =1

S(0, a)

e =1

S(0, –a)



Position of a Point with Respect to a Parabola

Let S y2 = 4ax be the equation of the parabola and P(x1, y1) be any point in the region of the parabola,then

S1 21y = 4ax1

If S1 > 0, then P lies outside the parabola.

If S1 = 0, then P lies on the parabola.If S1 < 0, then P lies inside the parabola.

Equation of a Chord

Let P( 21at , 2at1) and Q( 2

2at , 2at2) are two points on the parabola y2 = 4ax such that PQ is a focal chord.Then, the equation of chord is

y(t1 + t2) = 2x + 2at1t2

If the chord PQ is a focal chord of the parabola then (a, 0) must satify this equation

0 = 2a + 2at1t2 = t1t2 = –1

5

(e) Equation of directrix(f) Equation of axis(g) Vertex(h) Extremities of Latusrectum(i) Length of latusrectum(j) Equation of tangent at vertex

(k) Parametric equation

(l) Parametrix coordinates of anypoint on parabola

(m) Focal distance of any point P(h, k)on the parabola

(n) Equation of latusrectum

x + a = 0y = 0

A(0, 0)(a, + 2a)

4ax = 0

atyatx2

2

x – a = 0y = 0

A(0, 0)(–a, + 2a)

4ax = 0

atyatx

2

2

y + a = 0x = 0

A(0, 0)(+ 2a, a)

4ay = 0

2

2atyatx

y – a = 0x = 0

A(0, 0)(+ 2a, –a)

4ay = 0

2

2atyatx

P(at2, 2at)

h + a

x = a

P(–at2, 2at)

h – a

x + a = 0

P(2at, at2)

k + a

y = a

P(2at, –at2)

k – a

y + a = 0



Thus, if t is the parameter for one end of a focal chord, then parameter for other end is t1

and the

coordinates of the end points of a focal chord PQ of the parabola y2 = 4ax can be taken as P(at2, 2at) and Q

ta

ta 2,2 .

Length of Focal ChordLet P(at2, 2at) be the one end of a focal chord PQ of the parabola y2 = 2ax. Then, the lengths of focal

chord is a2

tlt , where t is the parameter for one end of the chord.

If l1 and l2 are two lengths of focal segments.

Then,21

214ll

lla

ie, length of latusrectum = 2

(Harmonic mean of the focal segments)

Intersection of a Line and a Parabola

Let the parabola be y2 = 4ax ...(i)

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

On eliminating x from Eqs. (i) and (ii), then

y2 = 4a

mcy

my2 – 4ay + 4ac = 0 ...(iii)

It is a quadratic equation in y.

Discriminant, D = (–4a)2 – 4(4ac)m = 16a(a – cm)

Now, If D > 0 ie, a > cm, then line intersect the parabola at two distnict points.

If D > 0 ie, a = cm, then line intersect the parabola at two coincident points ie, at one point.

If D < 0 ie, a < cm, then line neither touch nor intersect the parabola.

Condition of Tangency

The line y = mx + c will touch the parabola y2 = 4ax, if

mac

and point of contact will be

)0(,2,2

m

ma

ma

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Tangent to a Parabola

Equation of Tangent in Different Form

1. Point Form

The equation of tangent to the parabola y2 = 4ax at the point (x1, y1) is yy1 = 2a(x + x1). The equationof tangents to all standard forms of parabola at point (x1, y1) are given below.

2. Slop Form

The equation of tangent to the parabola y2 = 4ax in slop form is y = mx + ma

, where m is the slope

of the tangent.

The equation of tangent of slope m to the parabola (y – k)2 = 4a(x – h) is given by y – k = m(x – h)

+ ma

and coordinates of point of contact are

mak

mah 2,2 .

3. Parametric Form

The equation of tangent at the point (at2, 2at) or t to the parabola y2 = 4ax is ty = x + at2. Theparametric equation of tangents to all standard forms of parabola are given below.

Point of intersetion of Tangents at Any Two Points on the Parabola

Let the parabola be y2 = 4ax. Let two points on the parabola are P ( 21at , 2at1) and Q ( 2

2at , 2at2).

Equation of tangents at P( 21at , 2at1) and Q( 2

2at , 2at2) are t1y = x + 21at and t2y = x + 2

2at

Solving these equation, we get

x = at1t2, y = a(t1 + t2)

Thus, the coordingates of the point of intersection of tangents at ( 21at , 2at1) and ( 2

2at , 2at2) are

(at1t2, a(t1 + t2)).

yy1 = –2a(x + x1)

xx1 = 2a(y + y1)

xx1 = –2a(y + y1)

Equation of tangentEquation of parabola

y2 = 4ax

x2 = 4ay

x2 = –4ay

ty = –x + at2

tx = y + at2

tx = –y + at2

Equation of tangentEquation ofparabola

y2 = 4ax

x2 = 4ay

x2 = –4ay

Point of contact

(–at2, 2at)

(2at, at2)

(2at, –at2)

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Pair of Tangents

Let P(x1, y1) be any point outside the parabola S = y2 – 4ax, then the equation of the pair of tangents ofthe pair of tangents drawn from the point P to the parabola is given by

where, S = y2 – 4ax, S1 = 21y – 4ax1

and T = yy1 – 2a(x + x1)

Results on Tangents

1. The tangent at any point on a parabola bisects the angle between the focal distance of the point andthe perpendicular on the directrix from te point.

2. The tangents at the extremities of a focal chord of a parabola intersect at right angle on the directrix.

3. The protion ot the tangents to a parabola cut off between the directrix and the curve subtends a rightangle at the focus.

4. The perpendicular drawn from the focus on any tangent to a parabola intersect it at the point whereit cuts the tangents at the vertex.

5. The orthocentre of any triangle formed by three tangents to a parabola lies on the directrix.

6. The tangents at any point of a parabola is equallly inclined to the focal distance of the point and theaxis of the parabola.

Normal to a Paraboal

Equation of Tangent in Different Form

1. Point Form

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The equation of normal at the point (x1, y1) to the parabola y2 = 4ax is

(y – y1) = ay2

1 (x – x1)

2. Slop Form

The equation of normal to the parabola y2 = 4ax in the slope form is

y = mx – 2am – am3

The equation of normals to various standard forms of the parabola in terms of the slope of thenormal are as given below.

3. Parametric Form

The equation of normal at the point (at2 , 2at) or t to the parabola y2 = 4ax is y + tx = 2at + at3.

Point of Intersection of Normals Drawn Any Two Points on the Parabola

If two normals are drawn to the parabola y2 = 4ax at the points P( 21at , 2at1) and Q( 2

2at , 2at2) intersect at

a point R, then coordinates of the point R are {2a + a( 21t + 2

2t +t1t2), –at1t2(t1 + t2)}.

Results on Normals

1. The normal drawn at a point P( 21at , 2at1) to the parabola y2 = 4ax meets again the parabola at

Q( 22at , 2at2), then t2 = –t1 –

1

2t

.

2. The tangent at on extremity of the focal chord of a parabola is parallel to the normal at the otherextremity.

3. If the normals at point P( 21at , 2at1) and Q( 2

2at , 2at2) on the parabola y2 = 4ax meet on the parabola,then t1t2 = 2.

Equation ofparabola

y2 = –4ax

x2 = 4ay

x2 = –4ay

Equation of normal

y = mx + 2am + am3

x = my – 2am – am3

x = my + 2am + am3

Slope ofnormal

m

1/m

1/m

Point ofcontact

(–am2, 2am)

(–2am, am2)

(2am, –am2)

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4. If the normal chord at a point P( 21at , 2at1) to the parabola y2 = 4ax subtends a right angle at the

vertex of the parabola, then t2 = 2.

5. The normal chord of a parabolaat a point whose ordinate is equal to the abscissa, subtends a rightangle at the focus.

6. The normal at any point of a parabola is equally inclined to the focal distane to the point and theaxis of the parabola.

Conormal Points

The points on the parabola through which normals drawn are concurrent ie, pass through the same pointare called conormal points. The conormal points are aslo called the feet of the nromals. Points A, B, C inwhich the three normals from P(h, k) meet the parabola are called conormal points.

Results on Conormal Points

1. The algebraic sum of the slopes of the normals at conormal points is zero.

2. The sum of the ordinates of the conormal points is zero.

3. The centroid of the triangle formed by the conormal points on a parabola lies on its axis.

Chord of Contact

Let PQ and PR be tangents to the parabola y2 = 4ax drawn from any external point P(h, k),, the QR iscalled chord of contact of the parabola y2 = 4ax.

Let Q (x1, y1) and R (x2, y2)

Equation of the tangent PQ is

yy1 = 2a (x + x1) ...(i)

and equation of the tangent PR is

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yy2 = 2a (x + x2) ...(ii)

Since, lines (i) and (ii) pass through (h, k), then

ky1 = 2a(h + x1) ...(iii)

ky2 = 2a(h + x2) ...(iv)

Hence, it is clear Q(x1, y1) and R(x2, y2) lie on yk = 2a(x + h) which is chord of contact QR.

Equation of the Chord Bisected at a Given Point

The equation of the chord of the parabola y2 = 4ax which is bisected at (x1, y1) is

yy1 – 2a(x + x1) = – 4ax1

or T = S1

Where, T = yy1 – 2a(x – x1) and S1 = y2 – 4ax1.

Reflection Property of a Parabola

The tangent (PT) and normal (PN) of the parabola y2 = 4ax at P are the internal and external bisectorsof SPM and BP is parallel to the axis of the parabola and BPN = SPN.

All rays of light coming from the positive direction of x-axis and parallel to the axis of parabola afterreflection pass through the focus of the parabola.

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12

Note :

If the vertex of the parabola is (h, k), then generalised form of the parabola is (y – k)2 = 4a(x – h)2. Itsfocus is at (a + h, k) equation of directrix is x +a –h =0 and parametric equation are x =h +at2, y =k +2at

The length of focal chord having parameters t1 and t2 for its end point is a(t1 – t2)2.

The equation of tangent at (x1, y1) to any second degree curve can also be obtained by replacing x2 by

xx1, y2 by yy1, x by 2

1xx , y by 2

1yy and xy by 2

11 yxxy and without changing the constant (if any)

in the equation of the curve.

x-coordinate, at1t2 = GM of 21at , 2

2at .

y-coordinate, at1t2 = GM of 21at , 2

2at .

The line y = mx + c will be a normal to the parabola y2 = 4ax, if c = –2am – am3 and point of contactare (am2, –2am).

ram rajya more, siwan xi th , xii th , target iit-jee (main +...

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