# curve tracing 2.pdf

## Full text

(1)

### 1. Introduction

For evaluating areas, volumes

### of

revolution etc. we need to know the general nature of the given curve. In this

h

### apter

we shall learn the methods of tracing a curve in general and the properties

### of

some standard curves commonly met in engineering problems.

### I.

Symetry : Find out whether the curve is symmetri

### c

al about any line with the help of the following rules :

### ( 1)

The curve is symmetrical about the x -axis if the equati

### o

n of the curve remains unchanged when y is replaced

### by-

y i.e. if the equation contains only even powers of y.

### (2)

The curve is symmetrical about they-axis if the equation of the curve remains unchanged when x is replaced by-x

### i.e.

if the equation contains only even powers of x.

### (3)

The curve is symmetrical in opposite quadrants if the equation of the curve remains unchanged when

x andy are

e

ac

x and -y.

### (4) The

curve is symmetrical about the line y = x if the equation

### of

the curve remains unchanged when x andy

n

ang

.

II.

Find out whe

### ther the origin

lies on the curve.

### If it

does, find the equations of the tangents at the o

### rigin by

equating to zero the lowest degre terms.

III. Intersection

axes

### :

Find out the points

### of

intersection of the curve with the

### coordinate

axes. Find also the equations of

### the

tangents at these points.

Asymptotes

Find out the

totes if any.

n

### o

part of the curve lies

### :

Find out the regions of the

lane where no

### part

of the curve lies.

out dyldx

### :

Find out dy/dx and the points where the tangents

### are

parallel to the coordinate axes.

(2)

Mathemat - If

of Corwes

### studied

quite in detail straight line, circle, parabola,

an

inc

u

ng

hy

### perbo

la. We shall briefly review

curves.

### (a) Straight Line:

General equation of straight line is

the

ax +

+ c =

### 0.

To plot a straight line we put y =

and find x.

ls

we put x =

### 0

and find

y. We plot these two points

join them

### to

get the required line. Some particular cases of straight Jines are shown below. (i) Lines parallel to

axes.

y

x=h

0 X

Fig.

### (7.1)

(ii) Lines passing through origin y y=mx m>O X y y=k 0 X y

y=mx m<O X Fig.

Fig.

### (7 .4)

(iii) Lines making given intercepts on coordinate axes. y

X X

Fig.

Fig.

.

circle is

+

+

+

+ c =

### 0.

Its centre is

(3)

Engineering Mathematics

II

### (7- 3)

Tracing of Corves

### j2-

c. are shown below.

y

x-

### axis and passing through origin.

Its equation is of the form x2 +

±

=

y

-x

0

### (iv) Circle with centre on they-axis and passing through origin.

Its equation is of the form x2 +

±

=

y

0

### (v) Circle with centre on the axes but not passing through origin.

Its equation is of the form x2 +

±

+ c =

+

±

### 2fy

+ c ='

y

y

(4)

v- ..,, JTat:mg or �.;orves

Circle

th axes.

### Its

equation is of the form

+

±

= 0.

y

Fig.

### Ex. 1

: Draw the circle

+

+

### 4y + 4

= 0 Sol. : We write the equation as

+

+

2 =

Its centre

s

### 2.

It is shown on the right.

### Polar Coordinates

(a) Straight line Fig.

### (7.16)

If We put X = r COS 8, y = r Sin

### 8

in the equation Of the Straight line

ax+by+c=O

we get,

cos

+

sin

### e

+ c = 0 i.e. a cos

+

sin

+

= 0

r (i)

### A

line parallel to the initial line.

If a line is parallal to the initial line (or the

### x-

ax is) at a ditance p from it then, from the figure we see that its equation is p r sin

= ± p (It

clear that r sin

### e

= p, if the line is above the initial line and r sin

### e

=- p, if the line is 0 Fig.

### (7.17)

below the initial line.)

(ii)

### A

line perpendicular to the initial line. If a line is perpendienlar to the initial line (or the

x-axis) at a ditance p from it then from the figure we

see thatrcos6=±p

### (It

is clear that r cos

= p, if the line

### is

to the right 0 of the pole and r cos

=

### -

p, if the line is to the left of the pole.)

X

Fig.

. 0 p

X (iii)

### A

line through the pole (or the

### origin)

(5)

Engineering Mathematics

II

### (7- 5)

Tracing of Corves

a

g

e

### a

with the initalline then every point on the

has coordnates

### (r, a),

where r is positive or negative. Hence the e

u

t

on of a line is

a

a line

an • angle of

is

=

we put

cos

y

s

in the eq

at

### i

on of the circle with center at the o

x 2 + y 2 =

get

r 2

a 2

We put X =

COS

y

### = r

sin 0 in the equation

of the circle with center

X

2 + y 2

2 i.e.

x 2 + y

0

X

we get r 2

cos

0:.

cos

### e

This is the equation of the circle with ce

ter

### (a,

0)

and radius a in polar coordniates.

If we put X

cos

y

sin

the

### Fig. (7.21)

equation of the circle with center (0, a) and radius a i.e. in

x 2 +

a)

### 2 =a

2 i.e. x 2 + y 2-

0 we get, r 2

sin

= i.e. r =

### 2a sin e

This is the equation of the circle with center (0, a)

y and radius a in polar

coordniates

y

X

### Fig. (7.22)

If We put X= r COS 0, y = r sin 0 in the equation

of the cricle with center (a, a) and t

uchi

### ng

both the coordniate axes in the first quadrant

X i.e. in

a)2 +

a)2

a2

i.e. in

+

2ay + a2

### = 0

we get the equation r 2- 2ar. (cos

+sin 0) +a 2

### = 0

in polar coordniates.

### Ex.

: Find the equation of the cricle in polar coordinates having centre

### 3.

Also find the polar coordinates of its point of intersection with the line y =

### Sol.

: The equation of the circle with centre

### (3,

0) and

(6)

., --- ---:--- -- ,. - ..,, ••..•••11 WI �va-.aa

i.e. x2

### +y2-6x=O

Putting X = r COS

y = r Sin

we g

r 2-

cos

0 i.e. r

cos

y

x.

Putting X

r COS

= r s

n

we get the

equation in

coor

in

te

as

r sin

= r cos

1 :.

1t /4

Putting

=

r =

cos

, we get r

cos 1t

· =

J2

Hence

### the

polar coordinates of the point of intersection are

.1t

### Parabola :

General equation of parabola is

= ax

bx

or x

### ay 2 + by + c.

By completing the square on x or on

### y

and by shifting the origin the

equation can be written in standard form as

or x 2

4

d

x = a x

### Fig. (7.26)

The equation of the parabola with vertex at

and

### (i)

the axis parallel to the x-axis, opening to the right is

= 4

### (ii)

the axis parallel to the x-axis, opening to the left is

4

(x-

### h)

(iii) the axis parallel to the y-axis, opening upwards is (x-

=

### k)

(iv) the axis parallel to the y-axis, opening downwards is

=-4a

Sketch the curve

4x- 4y

0

The equation can

written as

4y + 4 =-4x-

i.e.

2 =-4 (x

### 1)

y

(7)

Engineering Mathematics -II

Putting y -2 = Y, x + 1 = X,

Its equation is Y 2 =

4X.

### (7- 7)

Tracing of Corves

Its vertex is at

### ( -1,

2) and it opens on the left.

### Ex. 2

: Sketch the curve

x2 + 4x- 4y +

=

### Sol.

: The equation can be written as

(x + 2)2 = 4

Putting x + 2 =

y -

### 3

= Y,

Its .equation is X 2 = 4

### Y.

Its vertex is (-2,

### 3)

and it opens upwards.

### (iv) Ellipse

The equation of ellipse in standard form is

1

=I (a<b) y y a>b

### (7.30)

If the centre of the ellipse is at (h, k) the equation of the ellipse becomes (x-

2

2 _

+

### b2

-1

If a> b, the major axis is parallel to the x-axis and if

### a<

b the major axis is parallel to the y-axis.

(8)

Mathematics -II

### (7- 8)

Tracing of Corves

y

y

### (vi) Rectangular Hyperbola

If the coordinate axes are the asymptotes the equation of the rectangular hyperbola is xy =

or xy = -

xy= k xy�-k

### Fig. (7.34)

If the lines y = + x and y = -x are the asymptotes, the equations of the rectangular hyperbola are given by

### -l

= a2, y2 -x2 = a2•

y X x2-y2=a2

' ' y2 -x2 = a2

### Fig. (7.36)

X

Polar equation of the rectangular hyperbola x2 -

### l

= a2 is obtained

by putting X = r cos

y = r sin

The equation is

cos2

- r2 sin2

= a2

r2 (cos2

sin2 8) = a2

r2 cos 2

### e

= a2

Similarly the polar equation of

### the

other rectangular hyperbola

### i.e. of

(9)

Engineering Mathematics -II

### (7· 9)

Tracing of Corves

### Ellipse

The parametric equations of

el

### li

pse are X = a cos

y =

sin

### Hyperbola

The parametric equations of

hyperbola are

X= a sec

y =

tan

or X= a

1, y =

### b sinh

1.

Parabola

The parametric equations of the

=

are x =a

, y =

### The

parametric equations of the parabola x2 = 4ay are x =

y =a

### ?

.

Rectangular Hyperbola

The parametric equations of the rectangular hyperbola xy = c2 are x = ct and y =

### Curves of

the form ym = xm where m

n are

### positive intigers.

It is interesting to note the shape of the curves whose equations can

### be

expressed in the above form.

### S

ome of the curves are known to us and some will be studied in the following pages. We give below the equations and the graphs of these curves.

y y

### (1)

y =x.

The staight line through the origin y

2

### (2)

y =x .

The parabola through the origin and opening

y

(10)

Mat.tMma -II

X

of two

### l

ines passing through the

i

in

y

=x.

The

la through

on

### i=�.

Semi-cubical parabola X X X

y3

Cubical parabola

### i =x2.

Semi-cubical parabola

:as oN :II

)(

--2 ,

--2

the x-

(ii)

curve passes

g

the origin and

x

tangents

o

n

=

the

### is a

double tangent at

origin. 3

(iii) Since

-x as

y � 0<\

the line x =

is an asymptote.

(iv)

h

n

>

is

the

not exist when

>

### (Remark

:By putting X= rcos

= r sine,

p

lar equation

(11)

-II

### (7- 11)

Tracing of Corves

Trace the curve

2

y3

y)

Sol.: (i)

### The

curve is symmetrical about they-axis. (ii) It passes through

### the

origin and the x-axis is a tangent at origin.

(iii) It meets the y-axis at

and

Further d y

dx. is zero at

### 2a);

the tangent at this point is parallel to the x-axis.

(iv) Since x2

y3

- y)

when y

and y <

### 0,

x2 is negative and the curve does not

exist for y >

and y <

Fig.

y

### Ex. 3

: Astroid or Four Cusped

Hypocycloid : Trace the curve

213 +

213

1

X= a

os

sin3

ol

### . : (i)

The curve is symmetrical about both axes.

### (ii)

The curve cuts the x-axis in

and

axis in

Fig.

### (7 .47)

(iii) Neither x can be greater than a

### nor

y

can be greater than

### Ex. 4:

Witch of Agnesi: Trace the curve xl = a2 (.rz-

y

: (i)

is symmetrical

the x-axis. (ii)

.

. . .

2 2

, v = a

cannot be

X '

negative. Also

cannot be

than a.

As

### x � 0, y�

oo, they-axis is an

sym

tote.

Trace

=

(x-

whete a,

: We consider

cases : (a) Case

: a

the x-axis.

x-axis

n

and

W

n x

### l>O

(12)

Engineering Mathematics

II

### (7- 12)

Tracing of Corves

when

x

c,

### l

is negative whenx>c,

### l >O

Hence, there is no curve to the left-of x =a and also between

=band

x=c.

### (4) If

x > c and increases then

also increases

y

x

a=

### <

c

y The �quation now becomes

=

x-

### ( 1)

As before the curve is symmetrical about

x the x-axis.

### (2)

It meets the x-axis in points

and

### , 0)

is an isolated point because if

x

x

### <c), l

is negative and if

x <

is negative.

### ( 4) If�

> c and increases

also increases.

<

### b

= c

y The equation now becomes

=

x

### (1)

The cmve is symmetrical about the

x-axis.

### (2)

It meets the x-axis in points

and

x

is negative.

If x

and increases,

### l.

also increases. y

=

= c

### (S.U. 2003)

The equation now becomes

= (x-

x

### (1)

As before the curve is symmetrical about the

- axis.

(13)

Mathematics

II

<a,

:

### curve

x112 + y112 = a112 Y

curve

y=x.

y =

= a112

:. 4x=a i.e.

curve

y-

:

+ x

are

d

re terms.

II >< Y

a,Z

��--� , __

2

2 X

tv

=

x =

y

x =

an

x

x <-a.

urv

curve :

+

y

### FJa. (7.55)

(14)

Engineering Mathematics -II

### (7- 14)

Tracing of Corves

y

'

:

= x2 (a +

x

-- x

left.

X

: y

: x

+ l) =

:. xi +

= ax2- x3 :.

=

2:

.\y2

=

The

-

... ,. " 2 •

be

x

x

### 3

+ y 3 = 3axy.

.. X

or x=--3at y=---3at2

l+t3, l+t3

x = y .

y

x

y

x

Ex. 4 :

x2

2-x2)

y �

X

x >

### a, l is negative., Hence there no curve beyond x = a,

x=-a.

(15)

Engineering Mathematics -II

Trace

he curvee

=

### b)2

Tracing of Corves

### Sol.:

We have already dicussed the above curve in Ex. L.ase III on page 7.12. The shap

### �

of the curve will remain unchanged

y replace y by

### .,Jc

. y i.e. the shape of the curve of the equation

=

will be the

same.

=

0.

Replacing c by

by zero and

by

### 3a,

we get the above curve. The curve is shown on the right.

y

2 t3

y

or

Replacing

by zero,

and

### c

1, we get the above curve. The curve is shown on the right.

=

Putting

= 0 and

=

### a

we get the above

curve. The curve is shown on the right. When x <

### 0,

y is imaginary and there is

no curve for x < 0

When x -7 oo or x -7-oo, y -7 oo.

y

(16)

Engineering Mathematics

II

c

<

<

y

y

--7 oo

--7

oo, y --7 oo.

### (S.U.1984, 95)

Tracing of Corves

X

:

y

y

X

:

y

+

### (S.U. 1984, 99, 2004)

(17)

Engineering Mathematics -II

+

X

+

=

+

+

+

=

+

### a2b2x2

Tracing of Corves

13.

+

x5 +

=

x

y = -1 2

X

Ex.

=

### Fig. (7.67)

X

(18)

a.JI�UIIIOIIOIIII� IWIUioiiiiOIIIIWiol

., 1•

y

x

### Fig. (7.68)

\'- IUJ ··- .. ···

-· --·

y

X :x

II

y X

y x

y

a=

a=

a= a

0 X

y

x

### Fig. (7.74)

(19)

Engineering Mathematics -II

y

y �+, X

### (7- 19)

Tracing of Carves

Y

to

Ex.

I

y

y

y

### Fig. (7.84)

X

(20)

Engineering Mathematics

II

### (7- 20)

Tracing of Corves

y

Same as 17

Similar to Ex.

### (30)

Similar to Ex. 17.

y X

### I.

Find out the symmetry using the following rules.

( 1) If on replacing

by

### - e

the equation of the curve remains unchanged then the curve is symmetrical about the initial line.

If on replacing

### r

by -r i.e. if the powers of r are even then the curve

is symmetrical about the pole. The pole is then called the centre of the curve.

### II.

Form the table of values of

### r

for both positive and negtive values of

### e.

Also find the values of

which give

and

.

Find tan

### «1>.

Also find the points where it is zero or infinity. Find the points at which the tangent coincides with the initial line or is perpendicular to it. (Refer to the explanation and the fig. in§

(b) page (2.11))

### IV.

Find out if the values of

and

### e

lie between certain limits i.e. find the greatest or least value of

### r

so as to see if the curve lies within or without a certain circle.

(a)

+cos

-cos

(c)

cos

### Sol. : 1 (a)

: (i) The curve is symmetrical about the initial line since its equation remains unchanged by replacing

by

y

X

r

1 + cos

### 6)

Fig. (7.88) (ii) When

and when

7t,

### r = 0.

(21)

Engineering Mathematics

II

Also when

### 8 =

rt/2, r = a and when

·

a

+cos

Now tan\$=

= -a sin

:. tan «l» =-cot

: . \$ = + Hence,

«l» gives

1t

1t

### '1'=8+2 +2 =2 +T

Tracing of Corves

When

=

### 0, 'I'

= i.e. the tangent at this point is perpendicular to the

line. When

1t

### 'I' = 2rt

i.e. the tangent at this point coincides with

### initial

line.

(iv) The following table gives some values of rand

### 1 (b)

: (i) The curve is symmetrical about the initial line.

Y (ii) When

=

r =

=

r=

when

1t, r=

r =a

cos

tan \$ = r

= a sin

r = a

1

cos

=tan

Hence, 'I'=

+ «l» gives

=

When

=

### 0, 'I'= 0

i.e. the tangent at this point coincides with the initial line when

= 1t,

=

### 31t/2

i.e. the tangent at this point is perpendicular to the initial line

### .

(iv) The following table gives some values of rand

: Squaring

=Vi cos (

we get r =a cos2

=

+cos

### 8).

This is the cardioide similar to the cardioide shown in

(a) with

### a/2

in place of a.

(22)

Engineering Mathematics -II

r =a+ b cos 0

### (a) Case

1 : a > b

(i) The curve is symmetrical about the initial line.

(ii) Since a > b

### ,

r is always

(iii) The following table gives some values of rand e. r

a +b

### (b) Case II

: a = b n/3 a+b/2 n/2 a

When a = b we get r = a (l +

### cos

0), the cardioide

in the

1 (i)

: a < b

(i) The curve

ymme

initial line.

(ii) r =

a + b cos 0 =

. I

z.e. e = cos -

### b .

Tracing of Corves X

2n/3 1t a--

a-b y a+b x

(iii) When 0 =

### 0,

r =a + b and is maximum.

(iv) When e = cos -1

-alb), r =

t

### h

e curve passes through

origin.

(v) When cos

### -I[-%]<

0 < n, r is negative and at 0 = n, r =a-- b.

L•

: It may be noted that to plot a point

- r, 0), we

3

### /

rotate the radius vector through 0 and r in

## [,'

o opposite direction in this position. For example, to get

the point(-

### 3, 45°),

we rotate the radius vector from OX

### through 45° into the

position OL. We have to

### measure

3 along OL a distance- 3 i.e. we have to measure a distance

### (7 92)

3 not along OL, but in opposite direction. Producing LO

P

### Ig.

toP so that OP = 3 units we get the required

### point

P. (vi) The following gives

### some

of rand 0

n/2 a

cos -1 (-alb) cos -1 (-a/b) < e < 1t

### 0

negative

1t a-b Note that we get a loop inside another toop as shown

### in

the figure.

(23)

Engineering Mathematics -II

### (7- 23)

Tracing of Corves

:

r =

+ 3cos

Y

### (S.U. 2007)

Sol. : Following the above line we see that the curve r = 2 + 3cos

### e

can be

shown as on the right. y

5

is also given by

r =

+

sin

### 8

where the y-axis becomes the

initial line.

For example, the curve r =

3sin

### 8

is

shown in the neighbouring figure.

:

=

cos

or

+

=

### (S.U.1995,97,98,2003)

9 = 37tl4 y 9 = 7tl4

r

### Sol.:

(i) Since on changing r by-r the equation

remains unchanged, the curve is symmetrical about the initial line.

(ii) Since on changing

by -

### 8

the

equation remains unchanged the curve is symmetrical about the pole.

(iii) Considering only positive values of

r we get the following table.

negative

imaginary

### a

(iv) Consider the equation

+

=

If we put from

i.e. y = ±

i.e.

=

and

= 3n

, we see that

=

and

=

### 3n/4

are tangents at the origin.

r =

sin

or r = a cos

: ( 1) Since sin

or cos

### n8

cannot be greater than one in both the cases

r cannot be greater than a. Hence, the curve wholly lies within the circle of

(24)

Engineering Mathematics -II

### (7- 24)

Tracing of Corves

### (2)

To find the curve r

sin

put

Then

### :. O=O, n, n

Draw these lines.

### If

n is odd there are

### n

loops in alternate divisions and if n is even there are

### 2n

loops one in each division.

### (3)

To trace the curve r

cos

put

Then,

....

### ' 2n ' 2n ' 2n

Draw these lines. If

is odd there are

### n

in alternate divisions and if

### n

is even there are 2n loops one in each division.

:

sin

(ii)

cos

9 =

--- --- 9 = n/3

### Sol.

: (i) The curve consists of three loops

lying within the circle

' '

' Put

:. sin

### 1t ' 21t ,

.... e ' ' ' ' ' '

### Fig. (7.96)

Draw these lines and place equal loops in alternate divisions.

9 =

### n/2

(ii) The curve consists of three loops

9 =

' ' ' ' ' ' 9 =

### Ex. 2

: Four leaved Rose

lying within the circle

Put

:. cos

### --6'6'2'6'6'2

Draw these lines and place equal loops in alternate divisions.

(i)

sin

(ii)

cos

### Sol.

: (i) The curve consists of four

x

### 2)

loops lying within the circle

. Put

sin

### = 0

(25)

Engineering Mathematics - II () = n/2 8 = Sn/4 ',___ __.-' () = -n/4 Fig. (7.99)

### (7- 25)

:. 20 = 0, n, 2n ....

:. 0 = 0,

### 2

' 1t , 2-Tracing of Corves

Draw these lines and place equal loops in all divisions.

(ii) The curve consists of four (2 x 2) loops lying within the circle

= a

Put r =

:. cos 20 =

1t

:. 20 =-

·

·

·

• ····

1t

:. 0=-

,

### 4 ' 4

' 0000

Draw these lines and place equal loops in all divisions.

### Ex. 3

: Spiral of Archimedes r = a 0

Sol. : Let us consider first only positive values of 0. When 0 = 0, r = 0, so the curve passes through

the pole. As 0 increases

, 1t ,

### 3

27t , 2n .. and so on we see that r increases and the point

tracing the curve goes away from the pole as shown

in the neighbouring figure. When 0 � oo, r � oo. Fig. (7.100)

Fig. (7.101) 0�=.

### r�O.

If we consider negative values of 0 also we get a curve shown by dotted line. (Fig.

### Ex.

4: Reciprocal Spiral

### r

0 =a (S.U. 1985)

Sol. : Let us first consider only positive values of

### e.

From the values considered below we see that as 0 increases r decreases. This means as 0 increases the point tracing the curve goes nearer and nearer to the

pole and when

1t

0 = ... 2n ,

n ,

_!I_ ..! 2a !I_

### 1E._

(26)

Engineering Mathematics -II

Fig.

### (7- 26)

Tracing of Corves

Further from the above values we note that as

-)

### 0,

r � oo . If we consider negative values

of 0 also we get a curve shown by dotted line. It may

### be

noted that the line y =

is an

asymptote.

Ex.

### 5

: Equiangular Spiral r

be

(S.U.

Sol. : When 0

, r

### = a.

As 0 increases r also

increases. When 0 -� oo , r � oo •

When 0 -> - 9 r ->

Further, tan

r

r r 1

:. tan

\

Fig.

Thus,

### <I> i.e.

the between the tangent and the radius vector is always constant. Hence, the name equiangular. Ex.

: r

(sec 0 + cos 0)

(S.U. 1979,

### 2004)

Sol. : The equaition can be written as

### [r

cos 0 r 2 =a (r sec

+ r cos 0) 0

X Fig.

### )

.. x +y +x :. J2 (x-

x

x)

### (i)

The curve is symetrical about x-axis. (ii) When x =

y =

+

=

### 0

are the tangents at the origin.

The tangents at

### the

origin are imaginary. The origin is an isolated point. (iii) x =

is an asymptote.

(a) r

+cos 0), (b) r

+cos 0 (S.U.

2. r

+ cos 0) (S.U.

### 1989)

(27)

Engineering Mathematics

11

1 +

cos

4 cos 2

l + cos

=

-cos

3 cos

(1 + sin

r

cos

### 8

Tracing of Corves

cos

or

=

sec

sin

(1 -sin

tan

sin

### 1.

Similar to solved Ex.l (1) Fig.

### 3.

Similar to solved Ex.

(a) Fig.

### 5.

Similar to solved Ex.

Fig.

### 2.

Similar to solved Ex. 1(1).

### 4.

Similar to solved Ex. 2 (c) Fig. (7.91)

### 6.

Parabola opening on the left.

Circle

### 7.

Parabola opening on the right.

### 9.

Cricle similar to the previous

one with a=

### 10.

y

(28)

Engineering Mathematics

II

### (7- 28)

Tracing of Corves

y

y

y

See remark under

X

Ex.

page (7

See Fig. (7

### (1) Cycloids

Because of its graceful form and beautiful properties, mathematicians call this curve the Helen of Geometry. (After all, mathematicians are not dry as they are sometimes called !) When a circle rolls, on a straight line without sliding any fixed point on its circumference traces a cycloid.

The curve traced by a fixed point on the cricumference of a cricle, which rolls without sliding on the cricumference of another fixed cricle is called an epicycloid or hypocycloid.

If the rolling circle is outside the fixed cricle the curve is called the

### epicycloid

and if the rolling cricle is inside the fixed cricle, the curve is called the

### hypocycloid.

We have seen one particular hypocycloid in Ex. (3) page

### 7.11.

If the radius of the rolling circle is equal to the radius of the fixed cricle, the epicycloid is called cardioide because of its heart like shape (Cardio

### =

Heart, eidos =shape). We have studied this curve Ex.

on page

### 7.20.

The cyloid is generally given in one of the following forms.

sin

- ccis

### (S.U. 2004, 07)

(29)

Engineering Mathematics

II

x �""a

y =a (1 -cost)

dx dy .

:

### (a) dt =a (1

+cost),

dt =-a sm t dy dy!dt --asint

### (t)

:. dx = dx/dt =a( 1 +cost)

=-tan

### Some of

the values of x, y and dyldx

-n -n/2 0

--a ( 1t/2 +

### ()

0 a 2a 00 1 0 Tracing of Corves

### (S.U. 1987)

n/2 n a(1t/2+1) an a 0 - 1 -

### oo

From the above table we see that at t =

### -

n and at t = n, the tangents are

parallel to they-axis. These points are called cusps.

at

= 0,

tangent

is parallel

the x-axis.

y t = 0

2a

dx dv

### (b)

-=a (l- cost) --=-a sin t

dt ' dt

dy dyldt --asint

: · dx =

d t = a ( 1

cos

=

### Some of

the values of x, y, dyldx are

0 n/2 n X 0 a (7tl2 -1) an y

a 0 dyldx -

### 00

-1 0 31t/2 2n a (3n/2 +1) 2an a 2a 1 00

From the above table we see that at t = 0 and t = 2n

### the

tangents are parallel to they-axis. These are cusps. Further at

### t

= n, y = 0 and dy/dx = 0 i.e.

the x-axis is the tangent.

t=O an: an:

---

### +

(30)

Engineering Mathematics -II

### (7-30)

Tracing of Corves

=a

=

t

_

### [t]

. - = ---- = -- tan

1

2

### Some of

the values of x, y,

are

-n:/2 ---

1t

+

_

y t = 1t : x

= a

t

=

cot

· ·

1 -cost) 2

### Some

of the values of x,

dyldx are

00

X

00 y

### a

1 t = n ' ' ' : 2a an : an

### 0

a -- 1 = 0 t:: 27t

-00

### (S.U. 2003)

If a heavy uniform string is allowed to hang freely under gravity, it hangs in the form a curve called catenary. Its equation can be shown to be

(31)

Engineering Mathematics

II

### (7- 31)

Tracing of Corves Its shape is as shown in the figure

y

\

### Fig. (7.117)

Its lowest point is at a dirtance c from the -cause,

when x =

y = c cosh

### 0

=c.

Further, we see that y

c

osh

### ;

= c

2

Hence, if x is replaced by - x the equation remains the same. Hence, the

y

_ e -xlc

Sm

e

### =

is zero. when x = c, the tangent at

### x

= c is

d.:r 2 parallel to the x-axis.:

Tractrix

x = a cos t +

a log tan 2

, y = a sin

a -=-asmt +­

2

sec2

tan 2

### [ f]

a = -a sin t + 2 sin

cos

= -a sin

+

=

- sin 2

sm t sm t

a cos 2

sin

=a

=

· ·

2

values

y,

### dx are ,

(32)

Engineering Mathematics - II

### (7- 32)

Tracing of Corves

t -n -n/2

n/2 1t

X 00

00

00

y

-a

a

dy/dx

-00

00

### 0

From the above data we see that as t �- n the point on the curve is (oo,

and as t �

### 0

the point is(- oo,

t = n/2

X

### Fig. (7.118)

You have studied to some extent plane and straight line

### in

three dimensions. We shall here get acquainted with some more three dimensional solids such as sphere, cylinder, cone, paraboloid etc. z

### 1. Plane

: The general equation of a plane is linear of the form a'x + b'y + c'z + d' =

### 0

which can be written as a' b' c' -x+-y+-z=1 d' -d'

### -d'

y=O X x=O i.e. ax + by + cz = 1

### Fig. (7.119)

y

Simplest planes are yz plane whose equation is x =

### 0;

the zx plane whose

is y =

### 0;

the xy plane whose equation is z =

z y

a a a y X X

x =±a, y = ±

z =±c.

cz = 1

### cuts

off intercepts 1/a, 1/b, 1/c on the

### axes. Still more common form of the plane is

�+l'.+�=l a b c

(33)

Engineering Mathematics -II

### (7- 33)

Tracing of Corves

This plane cuts off intercepts a, b, con the coordinate axes.

: An equation

### involving

only two variables represents a

cylinder in three dimensional geometry. Thus,

=

= 0, y (z,

### x)

= 0 represent

cylinders in three dimensions.

+

=

### a2

is a right circular cylinder, whose

generator is parallel to the z-axis. Similarly,

+

### z2:: !J2

is a cylinder whose generators are paralel to the x-axis. z2 +

### x2

= c2 is a cylinder x

### /

whose generators are parallel to they-axis.

z

..

X X

.

: The equation

### y2

= 4ax represents a paraola in

two dimensions. But in three dimensions it represents a parabolic cylinder.

The equations z2 =

### 4by,

x2 = 4cz also represent parabolic cylinders as

shown the following figures.

### 3. Sphere

: The equation of sphue in standard form i.e. with center at the origin and radius

is x2 + y2 +

= a2.

### 4. Cone

: The right circular cone given by

+

### y2

= z2 is shown in the

figure

7.129.

The other

y2 +

= x2 and x2

+

=

are shown in the

figures

and

### 7.31.

(34)

Engineering Mathematics -II

### (7- 34)

Tracing of Corves X 0

X

5.

### Ellipsoid

:The equation of ellipsoid in standard form is x2

### l

z2 -+-+-= 1 a2 b2 c2 z

### Fig. (7.131)

z

The sections of the ellipsoid by planes parallel to the coordinate planes are ellipses.

6.

### Hyperboloid

: The equations of

hyperboloid of one sheet and two sheet are

respectively

x2

z2 x2

### l

z2

-+-+-== 1 and -+-+-=-1.

a2 b2 c2 a2 b2 c2

The hyperboloid of one sheet is shown on the right. Sections of the hyperboloid of one sheet parallel to the xy-planes are ellipses and sections parallel to

### y

yz-plane or zx-plane are hyperbolas. x

z The hyperboloid of two

sheets

### is

shown on the left.

### Fig. (7.134)

Sections of this hyperboloid by planes paralles to the xy-plane i.e. z ==

### I

> c are real ellipses.

### Y

Sections of this hyperboloid by planes parallel to the zx-plane i.e. y =

are z

hyperbolas.

### 7. Paraboloids

: We shall discuss here only one elliptic paraboloid given by

x2

### l

2z

-+-==-a2 b2 c X

(35)

Engineering Mathematics -II

### (7- 35)

Tracing of Corves

Sections of this paraboloid parallel to the .xy-planes are ellipse. Sections parallel to the yz-plane or the zx-plane are parabolas.

Sketch the following figures. 2 2 2

### 1.

y=x ,x +y =2 2 2 2 2 2 X y

X + y = b ,

+

1 a b 7

### 3.

y- = X + 6, X = y

### 4.

y = x2, y = 1, X = 1, X = 2

### 5.

y=x2, y=x+2,x=-1,x=2

### 6

• x + y = 2 2 ax, ax = by 7. x2 +i=by,ax=by 2 2 2 2

### 8.

X - y = 1, X - y = 2, X)' = 4, xy = 2

xy = 2 - y

### 10.

y = x2 -3x, y = 2x

i =-4 (x-1),

=-2 (x-2)

a2 x2 = 4l (a2

### -l)

or x =a sin 2t, y =a sin t

### 13.

r (cos 8 +sin 8) = 2a sin 8 cos 8 or (x + y) (x2 +

= 2axy

### 14

. x + y = a xy or 4 4 2 2 r 2 = 2a2 sin8 cos8

4 4 sin 8 +cos 8 . 2 8 7 8

### 15

• x + y = a x y or 6 6 2 2 2 r 2 a 2 sm · cos-6 6 sin 8+ cos 8

x4-2.tya2+ a2l =

or

### ?

= 2a tan 8 sec2 8 - a2 tan 2 8 sec2 8

### 17.

y=x2 -6x+3, y=2x-9

### 18.

y = 3x2 - x-3, y =- 2x2 + 4x + 7

= 4x, y = 2x- 4

### 20.

y = 4x -x2, y = x

### 21.

A cylindrical hole in a sphere.

### 22.

A cone and paraboloid given b y z 2 = x2 +land z = x2 +

A paraboloid� +

### l

= az, and the c ylinder� +

= a2

2 2

### 24.

The paraboloids z 4 -x2 - Y4 , z = 3x2 + y4

(36)

Engineering Mathematics -II

y x2+y2=2

y

y

y

### Fig. (7.142)

X X X Tracing of Corves

Y x2+y2=b2 X

y=1 x=1 x=2

y X

y X

### Fig. (7.143)

(37)

Engineering Mathematics -II

### (7- 37)

Tracing of Corves

y

X

X

X

y

y X X

y

y X X

### Fig. (7.151)

(38)

Engineering Mathematics II

y

X

y

### (-1, 1)

Tracing of Corves y

### .159)

(39)

Engineering Mathematics -II

y

Fig.

### (7- 39)

Tracing of Corves

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## References

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