ASYMPTOTESASYMPTOTES

An An asymptoteasymptote of a of a curvecurve is a line is a line such that the distance between such that the distance between

the curve and the line approaches the curve and the line approaches zero as they tend to infinityzero as they tend to infinity

X

Y

TYPES OF TYPES OF ASYMPTOTESASYMPTOTES

ASYMPTOTES

VERTICALASYMPTOTES

HORZONTALASYMPTOTES

OBLIQUE ASYMPTOTES

VERTICAL ASYMPTOTESVERTICAL ASYMPTOTES

The line The line xx = = aa is a is a vertical asymptotevertical asymptote of the graph of the graph of the function of the function

yy = = ƒ ƒ ((xx) if at least one of the following ) if at least one of the following statements is true:statements is true:

Lim f (x) =+Lim f (x) =+∞ or -∞ or Lim f (x)=+∞ or -∞∞ or -∞ or Lim f (x)=+∞ or -∞

X → a+ X → a-

CURVE

Y

X

ASYMPTOTES

The function The function ƒƒ((xx) may or may not be defined at ) may or may not be defined at aa, and its precise value , and its precise value at the point at the point xx = = aa does not affect the asymptote. does not affect the asymptote.

For example, for the functionFor example, for the function

f(x)=1/x for x>0 f(x)=1/x for x>0 or 5 for x<=0 or 5 for x<=0

has a limit of +∞ as has a limit of +∞ as xx → 0+, → 0+, ƒƒ((xx) has the vertical asymptote ) has the vertical asymptote xx = 0, even = 0, even though though ƒƒ(0) = 5. The graph of this function does intersect the vertical (0) = 5. The graph of this function does intersect the vertical asymptote once, at (0,5). It is impossible for the graph of a function to asymptote once, at (0,5). It is impossible for the graph of a function to

intersect a vertical asymptote (or a vertical line in general) in more than intersect a vertical asymptote (or a vertical line in general) in more than one point.one point.

A common example of a vertical asymptote is the case of a rational A common example of a vertical asymptote is the case of a rational function at a point x such that the denominator is zero and the function at a point x such that the denominator is zero and the

numerator is non-zero.numerator is non-zero.

HORIZONTAL ASYMPTOTESHORIZONTAL ASYMPTOTES

Horizontal asymptotesHorizontal asymptotes are horizontal lines that are horizontal lines that the graph of the function approaches as the graph of the function approaches as xx → ±∞. → ±∞.

The horizontal line The horizontal line yy =  = cc is a horizontal is a horizontal asymptote of the function asymptote of the function yy =  = ƒƒ((xx) )

if Lim f(x)=cif Lim f(x)=c

or or Lim f(x)=cLim f(x)=c

In the first case, In the first case, ƒƒ((xx) has ) has yy =  = cc as asymptote as asymptote when when xx tends to −∞, and in the second that tends to −∞, and in the second that ƒƒ((xx) )

has has yy =  = cc as an asymptote as as an asymptote as xx tends to +∞ tends to +∞

x→∞

x→ -∞

curve

asymptotes

Y

X

OBLIQUE ASYMPTOTESOBLIQUE ASYMPTOTES

When a linear asymptote is not parallel to When a linear asymptote is not parallel to the the xx- or - or yy-axis, it is called an -axis, it is called an oblique oblique

asymptoteasymptote or or slant asymptoteslant asymptote. A function . A function ff((xx) is asymptotic to the straight line ) is asymptotic to the straight line yy = =

mxmx + + nn ( (mm ≠ 0) ≠ 0)

if Lim [f (x) - (mx+n) ]=0if Lim [f (x) - (mx+n) ]=0

OrOr

Lim [f (x) - (mx+n) ]=0Lim [f (x) - (mx+n) ]=0X → -∞

x→∞

m x +b

curve

asymptotes

Y

X

WORKING RULE TO FIND OBLIQUE WORKING RULE TO FIND OBLIQUE ASYMPTOTES OF AN ALGEBRAIC ASYMPTOTES OF AN ALGEBRAIC

CURVESCURVES

1.1. In the highest degree terms,put In the highest degree terms,put x=1,y=m and obtain x=1,y=m and obtain ΦΦ (m).Then in (m).Then in the next highest degree term again the next highest degree term again put x=1,y=m to obtain put x=1,y=m to obtain ΦΦ (m) and (m) and so on.so on.

2.2. Put Put ΦΦ (m)=0,then its roots say (m)=0,then its roots say mm11,m,m22…. are the slopes of the …. are the slopes of the asymptotes.asymptotes.

n

n-1

n

33.For the non repeated roots of .For the non repeated roots of ΦΦ (m)=0,find c from the relation (m)=0,find c from the relation

c=-c=-ΦΦ (m)/ (m)/ΦΦ (m) for each (m) for each valuevalue

of m.of m.

The required asymptotes areThe required asymptotes are

y=my=m11x+cx+c11

y=my=m22x+cx+c22

y=my=m33x+cx+c33 ………… …………

n

n-1 n

44.If .If ΦΦ (m)=0 for some value of m but (m)=0 for some value of m but

ΦΦ (m)≠0 ,then there is no asymptotes (m)≠0 ,then there is no asymptotes corresponding to that value of m.corresponding to that value of m.

55.If .If ΦΦ "'"'(m)=0 and also (m)=0 and also ΦΦ (m)=0 which (m)=0 which is the case when two of the is the case when two of the asymptotes are parallel, then find c asymptotes are parallel, then find c from the equationfrom the equation

(c²/2!)(c²/2!)ΦΦ (m)+ c (m)+ cΦΦ (m)+ (m)+ΦΦ (m)=0 (m)=0

which gives two values of c. Thus which gives two values of c. Thus there are two parallel asymptotes there are two parallel asymptotes corresponding to this value of m.corresponding to this value of m.

n

n-1

n

‘

n-1

n n-1 n-2

‘

′

″

′

66.If .If ΦΦ (m)= (m)=ΦΦ (m)= (m)=ΦΦ (m)=0,then (m)=0,then the values of c corresponding to this the values of c corresponding to this value of m are determined from the value of m are determined from the equationequation

(c³/3!)(c³/3!)ΦΦ (m)+(c²/2!) (m)+(c²/2!)ΦΦ (m)+c (m)+cΦΦ’’ (m)=0.(m)=0.

nn-1 n-2

n n-1 n-1

″

″′″

′

EXAMPLEEXAMPLE

TO FIND ASYMPTOTES OF TO FIND ASYMPTOTES OF ALGEBRAIC CURVE ALGEBRAIC CURVE

Find the asymptotes of the curveFind the asymptotes of the curvexx³³+3x+3x²²y-4yy-4y³³-x+y+3=0?-x+y+3=0?

SOLUTION: SOLUTION: The given curve is The given curve is xx³³+3x+3x²²y-4yy-4y³³-x+y+3=0-x+y+3=0 To findTo find the oblique asymptotes the oblique asymptotes Putting x=1,y=m in the third, second and Putting x=1,y=m in the third, second and

first degree terms one by one, we getfirst degree terms one by one, we get

ΦΦ (m)=1+3-4m³ (m)=1+3-4m³

ΦΦ (m)=0 (m)=0

ΦΦ (m)=-1+m (m)=-1+m

NowNow ΦΦ (m)=0 (m)=0

1+3m-4m³=01+3m-4m³=0

(1-m)(1+4m+4m²)=0(1-m)(1+4m+4m²)=0

m=1,m=-½,m=-½m=1,m=-½,m=-½

Also Also ΦΦ (m)=3-12m² (m)=3-12m²

and and ΦΦ (m)=-24m (m)=-24m

3

2

1

3

3′

3″

therefore,therefore, c=- c=-ΦΦ (m)/ (m)/ΦΦ (m) (m)

=-=-ΦΦ (m)/ (m)/ΦΦ (m) (m)

=-(0/(3-12m²))=-(0/(3-12m²))

WhenWhen m=1, c=-(0/(3-12))=0 m=1, c=-(0/(3-12))=0

WhenWhen m=-1/2 m=-1/2

c=-(0/(3-3))=0c=-(0/(3-3))=0

ThereforeTherefore in this case, c is given by in this case, c is given by

(c²/2!)(c²/2!)ΦΦ (m)+c (m)+cΦΦ (m)+ (m)+ΦΦ(m)=0(m)=0

(c²/2!)(-24m)+c.0+(m-1)=0(c²/2!)(-24m)+c.0+(m-1)=0

n-1 n′

2 3′

3″

2′

1

(c(c²²/2)(12)-(1/2)-1=0/2)(12)-(1/2)-1=0

cc²=1/4²=1/4

c=±1/2, for m=-1/2c=±1/2, for m=-1/2

NowNow the asymptotes are obtained by the asymptotes are obtained by putting the values of m and c in putting the values of m and c in y=mx+c,y=mx+c,

i.e. y=x+0 ,i.e. y=x+0 ,

y=(-1/2)x+1/2y=(-1/2)x+1/2

and y=(-1/2)x+(-1/2)and y=(-1/2)x+(-1/2)

WORKING RULE FOR FINDING WORKING RULE FOR FINDING ASYMPTOTES OF POLAR ASYMPTOTES OF POLAR

CURVESCURVES

11.In the given equation ,put r=1/u..In the given equation ,put r=1/u.

2.2.If the given equation involves If the given equation involves trigonometric ratios, convert them to trigonometric ratios, convert them to sinsinθθ and cos and cosθθ..

find the limit of find the limit of θθ as u as u →→ 0. 0.

Let Let θθ , ,θθ ,….., ,…..,θθ be the limits obtained. be the limits obtained.

3.3. Find p=lim(-d Find p=lim(-dθθ/du) for each of the /du) for each of the θθ , ,

obtained in step 2.obtained in step 2.

44.The corresponding asymptotes are .The corresponding asymptotes are given by the equation p=r sin(given by the equation p=r sin(θθ--θθ ). ).

1 2 n

θ θ

u 0

i

i

iii

EXAMPLE EXAMPLE

TO FIND ASYMPTOTES OF TO FIND ASYMPTOTES OF

POLAR CURVEPOLAR CURVE

1.FIND THE ASYMPTOTES OF 1.FIND THE ASYMPTOTES OF THE CURVE r=(2a)/(1+2cosTHE CURVE r=(2a)/(1+2cosθθ)?)?

SOLUTION: SOLUTION: Putting u=1/r,the equation of Putting u=1/r,the equation of

the curve becomesthe curve becomes 2au=1+2cos2au=1+2cosθθWhenWhen u u →→ 0,then 0,then coscosθθ =-1/2 =-1/2 = cos2= cos2ππ/3/3 θθ =2n =2nππ±2±2ππ/3/3

1

1

DifferentiatingDifferentiating(1) w.r.t (1) w.r.t θθ, we have, we have

2a(du/d2a(du/dθθ)=-2sin)=-2sinθθ

or dor dθθ/du=-a cosec/du=-a cosecθθ

p= lim(-dp= lim(-dθθ/du) /du)

= lim a cosec= lim a cosecθθ

=a cosec(2n=a cosec(2nππ±2±2ππ/3)/3)

θ→θ1

==±a cosec2±a cosec2ππ/3/3 =±2a/√3=±2a/√3The asymptotesThe asymptotes is is p=r sin(p=r sin(θθ – –θθ ) ) ±2a/√3 =r sin[2n±2a/√3 =r sin[2nππ±2±2ππ/3-/3-θθ] ]

±2a/√3 =r sin[±2±2a/√3 =r sin[±2ππ/3-/3-θθ]] =r sin[2=r sin[2ππ/3±/3±θθ]]

1

2. Find the asymptotes of the curve 2. Find the asymptotes of the curve r=a secr=a secθθ+b tan+b tanθθ??

SOLUTION:SOLUTION: The given curve isThe given curve is

r=a/cosr=a/cosθθ+b (sin+b (sinθθ/cos/cosθθ

=(a+b sin=(a+b sinθθ)/cos)/cosθθ

u=cosu=cosθθ/(a+bsin/(a+bsinθθ) ,where u=1/r ..1 ) ,where u=1/r ..1

du/ddu/dθθ=(a+bsin=(a+bsinθθ)(-sin)(-sinθθ)-(cos)-(cosθθ)(bcos)(bcosθθ) .2) .2

(a+bsin(a+bsinθθ) )

=-(a sin=-(a sinθθ+b)/(a+b sin+b)/(a+b sinθθ))²²

From(1)From(1), u 0 gives cos, u 0 gives cosθθ →→ 0 0

i.e. i.e. θθ =(2n+1) =(2n+1)ππ/2/2

NowNow, p=lim(-d, p=lim(-dθθ/du) /du)

=lim(a+bsin=lim(a+bsinθθ)²/(asin)²/(asinθθ+b)+b)

=[a+bsin(2n+1)=[a+bsin(2n+1)ππ/2]²/2]²

asin(2n+1)asin(2n+1)ππ/2+b /2+b

1

θ → θi

=[a+b(-1)=[a+b(-1)ⁿsinⁿsinππ/2]²/2]²

a(-1)ⁿsina(-1)ⁿsinππ/2+b/2+b

=[a+b(-1)ⁿ]²=[a+b(-1)ⁿ]²

a(-1)ⁿ+ba(-1)ⁿ+b

The equation ofThe equation of asymptotes asymptotes is is

p=r sin(p=r sin(θθ--θθ ) )

[a+b(-1)ⁿ]² =[a+b(-1)ⁿ]² =r sin[(2n+1)r sin[(2n+1)ππ/2-/2-θθ

a(-1)ⁿ+b a(-1)ⁿ+b

=(-1)ⁿ r cos=(-1)ⁿ r cosθθ

1

CURVE CURVE TRACINGTRACING

METHOD METHOD

OF TRACING A CURVEOF TRACING A CURVE

1.Symmetry  (i) A curve is symmetrical about x-axis if the equation remains the  same by replacing y by            –y. here y should have even powers only.             For example y2= 4ax.  (ii) It is symmetrical about y-axis if it contains only even powers of x  For example x2 = 4ay (iii) If on interchanging x and y, the equation remains the same then  the curve is           symmetrical about the line. y = x,     For example x3 + y3= 3axy

(iv) A curve is symmetrical in the opposite quadrants if its equation remains the                same when x and y replaced by –x and –y.  For example y = x3 

Y

XO

Y’

Y

XX’O

Y

Y’

XX’

O

Y

X

Y’

X’O

The curve passes through the origin, if the equation does not contain

constant term.

For example the curve y2 = 4ax passes through the origin.

2. (a) Origin

(b)(b) Tangents at the origin:Tangents at the origin:

To know the nature of a multiple To know the nature of a multiple point it necessary to find the tangent point it necessary to find the tangent at that point. at that point. The equation of the tangent at the The equation of the tangent at the origin can be obtained by equating to origin can be obtained by equating to zero, the lowest degree term in the zero, the lowest degree term in the equation of the curve. equation of the curve.

3. 3. AsymptotesAsymptotes

AsymptotesAsymptotes are the tangents to the are the tangents to the curve at infinity. curve at infinity. 

(a)(a) Asymptote parallel to the x-Asymptote parallel to the x-axisaxis is obtained by equating to zero, the is obtained by equating to zero, the coefficient of the highest power of x.coefficient of the highest power of x.  For exampleFor example yx yx22 - 4x - 4x22 + x + 2 = 0   + x + 2 = 0           (y - 4)x         (y - 4)x22 + x + 2 = 0 + x + 2 = 0   The coefficient of the highest power of x    The coefficient of the highest power of x i.ei.e. x. x22 is y - 4 = 0 is y - 4 = 0          y - 4 = 0 is the asymptote parallel to           y - 4 = 0 is the asymptote parallel to the x axis. the x axis.

(b)(b) Asymptote parallel to theAsymptote parallel to the y-axisy-axis is obtained to zero, the coefficient of is obtained to zero, the coefficient of

highest power of y. highest power of y.  For exampleFor example

                                  xy                                  xy33 - 2y - 2y33 + y + y22 + x + x22 + 2 = 0 + 2 = 0                                    (x-2)y                                    (x-2)y33 + y + y22 + x + x22 + 2 = 0 + 2 = 0The coefficient of the highest power of y. i.e. The coefficient of the highest power of y. i.e.

yy33 is x -2. is x -2.X -2 = 0 is the asymptote parallel to y-axis. X -2 = 0 is the asymptote parallel to y-axis.

4. 4. The points of intersection with The points of intersection with the axesthe axes

(i)(i) By putting y= 0 in the equation of the By putting y= 0 in the equation of the curve we get the co-ordinates of the point curve we get the co-ordinates of the point of intersection with the x –axis. of intersection with the x –axis. 

  (ii)(ii) By putting x = 0 in the equation of the By putting x = 0 in the equation of the curve; the ordinate of the point of  curve; the ordinate of the point of  intersection with the y-axis is obtained by intersection with the y-axis is obtained by solving the new equation. solving the new equation. 

5. 5. Regions in which the curve Regions in which the curve does not liedoes not lie..

If the value of y is imaginary for If the value of y is imaginary for certain value of x then the curve certain value of x then the curve does not exist for such values.does not exist for such values.

ExampleExample. y. y22 = 4x   = 4x  For negative value of x, y is For negative value of x, y is imaginary so there is no curve is imaginary so there is no curve is second and third quadrant. second and third quadrant.

(i)(i) For y> 2a x is imaginary so For y> 2a x is imaginary so there is no curve in second and there is no curve in second and

third quadrant  third quadrant  (ii)(ii) For negative values of y, x is For negative values of y, x is imaginary. There is no curve in imaginary. There is no curve in

3rd and 4th quadrant. 3rd and 4th quadrant.

6. 6. SPECIAL POINTSSPECIAL POINTS

    Put 0=dx/dy for the points where tangent is Put 0=dx/dy for the points where tangent is parallel to the x-axis.parallel to the x-axis.  For exampleFor example x x22 + y + y22 - 4x + 4y -1 = 0 ………(1) - 4x + 4y -1 = 0 ………(1)                         2x +2ydxdy- 4 + 4dxdy= 0                         2x +2ydxdy- 4 + 4dxdy= 0                         (2y+4) dx/dy= 4 - 2x                          (2y+4) dx/dy= 4 - 2x

or dx/dy=(4-2x)/(2y+4) or dx/dy=(4-2x)/(2y+4)  NowNow dx/dy= 0. dx/dy= 0.

4-2x = 0 or x = 2 4-2x = 0 or x = 2 Putting x =2 in (1), Putting x =2 in (1),

we getwe get y2 + 4y -5 = 0  y2 + 4y -5 = 0                           y= 1,- 5                           y= 1,- 5 The tangents are parallel to x-axis at the points The tangents are parallel to x-axis at the points (2,1) and (2, -5). (2,1) and (2, -5).

This procedure can be This procedure can be remembered asremembered as : :

SOAP-RS

symmetry

origin asymptotes

Point of intersection

regionSpecial point

EXAMPLEEXAMPLE

1. 1. Trace the curve yTrace the curve y22( 2a – x ) = x( 2a – x ) = x33

Solution: y2= x3/( 2a – x ) ……..(1) 

(i) Symmetrical about x-axis:

Equation contains only even powers of y; therefore, it is symmetrical about x-axis. 

Y

Y=0

2a

O

CISSOID

ASYMPTOTES

TANGENT

X

Y’

(ii) Origin:

Equation does not contain any constant term;

therefore, it passes through origin. 

(iii) Asymptote parallel to y-axis:

Equation of asymptote is obtained by equating the coefficient of lowest degree of y. 

2ay2 – xy2 = x3 or (2a-x)y2 = x3

 Eq. Of asymptote is 2a-x = 0 or x = 2a. 

(iv) Region of absence of curve:

y2 becomes negative on putting x>2a or x<0, therefore, the curve does not exist for x<0 and x>2a.

(v) Tangent at the origin:

Equation of the tangent is obtained by equating to zero the lowest degree terms in the equation (1).           2ay2 – xy2 = x3           Equation of tangent:           2ay2 = 0 → y2 = 0   

2.Trace the curve 2.Trace the curve

xx22yy22=(a+y)=(a+y)22(a(a22-y-y22))

Proof:Proof:

The given equation of the curve The given equation of the curve isis

xx22yy22=(a+y)=(a+y)22(a(a22-y-y22))

yy22

Y

X

B(0,a)

B’(0,-a)

Y’

X’ O

Y=a

1.Symmetry:1.Symmetry:

Because of even powers of Because of even powers of x only, the curve is symmetrical x only, the curve is symmetrical about y-axis.about y-axis.

2. 2. originorigin::

The curve does not pass The curve does not pass through the origin. through the origin.

3.3.ASYMPTOTESASYMPTOTES::

The equation of the The equation of the curve is of 4curve is of 4thth degree in x and y. The degree in x and y. The

terms containing xterms containing x4 4 and xand x33 are are absent .The coefficient of xabsent .The coefficient of x22 is y is y2.2.

Therefore the asymptotes parallel to x-Therefore the asymptotes parallel to x-axis is y=0.There is no other axis is y=0.There is no other asymptotes to the curve.asymptotes to the curve.

4. 4. Points of intersection with co-Points of intersection with co-ordinate axis:ordinate axis:

The curve does not meet x-The curve does not meet x-axis.It meets y-axis,where axis.It meets y-axis,where x=0.Therefore y=-a,y=a.x=0.Therefore y=-a,y=a.

Shifting the origin at (0,a),the Shifting the origin at (0,a),the equation of the curve transforms intoequation of the curve transforms into

XX22(Y+a)(Y+a)22=(Y+2a)=(Y+2a)22(-Y)(-Y)

Thus the equation of the tangent at Thus the equation of the tangent at the new origin is Y=0 or y=a is the new origin is Y=0 or y=a is tangent at (0,a).tangent at (0,a).

Again shifting the origin at (0,-a),the Again shifting the origin at (0,-a),the equation transform into equation transform into

XX22(Y-a)(Y-a)22=Y=Y33(2a-Y).(2a-Y).

Therefore the tangents at the origin Therefore the tangents at the origin are given by Xare given by X22=0 i.e y-axis is a =0 i.e y-axis is a cuspidal tangent at (0,-a).cuspidal tangent at (0,-a).

5.5.RegionRegion::

The whole of the curve lies The whole of the curve lies between y=-a and y=a.between y=-a and y=a.

Hence the shape of the curve is as Hence the shape of the curve is as shown in figure.shown in figure.

ASSIGNMENTASSIGNMENT

11.If y=√logx+√logx+……..+∞, prove .If y=√logx+√logx+……..+∞, prove that (2y-1)dy=1that (2y-1)dy=1

dx xdx x

22.If f(x)=x³+2x²-5x+11,find the value .If f(x)=x³+2x²-5x+11,find the value of f(9/10) with the help of taylor’s of f(9/10) with the help of taylor’s series for f(x+h).series for f(x+h).

33.State and prove MACLAURIN’S .State and prove MACLAURIN’S theorem with Lagrange’s form of theorem with Lagrange’s form of remainder after n terms.remainder after n terms.

44.Find all the asymptotes of the curve .Find all the asymptotes of the curve x(y-x)²-x(y-x)-2=0.x(y-x)²-x(y-x)-2=0.

55.If .If 1 1 and and 22 are the radii of curvature are the radii of curvature at the extremities of a focal chord of at the extremities of a focal chord of a parabola whose semi-latus rectum a parabola whose semi-latus rectum is l,prove that (is l,prove that (11))-2/3-2/3+(+(22))-2/3-2/3=(l)=(l)-2/3.-2/3.

66.If c.If cx,c+y x,c+y be the chord of curvature be the chord of curvature parallel to coordinates axes at any parallel to coordinates axes at any point of the curve y=c cosh(x/c);point of the curve y=c cosh(x/c);

prove the 4cprove the 4c22(c(cxx2+2+ccyy

22)=c)=cyy4.4.

77.If y=(cos x).If y=(cos x)(cos x)(cos x)…∞(cos x)(cos x)…∞,prove that,prove that

dy=-ydy=-y22tan xtan x

dx 1-y log cos xdx 1-y log cos x

88.If y=(sin.If y=(sin-1-1x)x)22, prove that , prove that

(1-x(1-x22)y)yn+2n+2-(2n+1)xy-(2n+1)xyn+1n+1-n-n22yynn=0.=0.

99.Differentiate sin.Differentiate sin33x w.r.t. cosx w.r.t. cos33x.x.

1010.Evaluate lim x.Evaluate lim xbb-b-bx .x .

xxxx-b-bbbbx

TESTTEST

SET-ASET-A

Attempt any twoAttempt any two

11.Find the asymptotes of the hyperbola .Find the asymptotes of the hyperbola xx22-y-y2 2 =1=1

9 49 422.Find all the asymptotes of the following .Find all the asymptotes of the following

curvescurves x(y-x)x(y-x)22-x(y-x)=2-x(y-x)=233.Find the asymptotes of curve .Find the asymptotes of curve r cos 2r cos 2θθ=a sin 3=a sin 3θθ

SET-BSET-B

Attempt any twoAttempt any two

11.Find the equation of the hyperbola .Find the equation of the hyperbola having x+y-1=0 and x-y+2=0 as its having x+y-1=0 and x-y+2=0 as its asymptotes and passing through asymptotes and passing through the origin.the origin.

22.Find the asymptotes of the curve .Find the asymptotes of the curve r=a sec r=a sec θθ+b tan +b tan θθ

33.Find the asymptotes of y=x. e.Find the asymptotes of y=x. exx-e-e-x-x

eexx+e+e-x-x