B.Tech-II_Unit-I

Unit-1 CURVE TRACING

RAI UNIVERSITY, AHMEDABAD 1

Course: B.Tech- IISubject: Engineering Mathematics II

Unit-1RAI UNIVERSITY, AHMEDABAD

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Unit-I: CURVE TRACING

Sr. No. Name of the Topic Page No.

1 Important Definitions 2

2 Method of Tracing Curve 3

3 Examples 7

4 Some Important Curves 11

5 Exercise 13

6 Reference Book 13




CURVE TRACING

INTRODUCTION: The knowledge of curve tracing is to avoid the labour of plotting a large number of points. It is helpful in finding the length of curve, area, volume and surface area. The limits of integration can be easily found on tracing the curve roughly.

1.1 IMPORTANT DEFINITIONS:

(I) Singular Points: This is an unusual point on a curve.(II) Multiple points: A point through which a curve passes more than one

time.(III) A double Point: If a curve passes two times through a point, then this

point is called a double point.(a) Node: A double point at which two real tangents (not coincident) can be drawn.(b) Cusp: A double point is called cusp if the two tangents at it are coincident.

(IV) Point of inflexion: A point where the curve crosses the tangent is called a point of inflexion.

(V) Conjugate point: This is an isolated point. In its neighbour there is no real point of the curve.

At each double point of the curve y=f(x), we get,




= = ×a) If D is +ve, double point is a node or conjugate point.b) If D is 0, double point is a cusp or conjugate point.c) If D is –ve, double point is a conjugate point.

2.1 METHOD OF TRACING A CURVE:

This method is used in Cartesian Equation.

1. Symmetry: (a) A curve is symmetric about x-axis if the equation remains the same by

replacing y by –y. here y should have even powers only.For ex: =4ax.

(b) It is symmetric about y-axis if it contains only even powers of x.For ex: =4ay

(c) If on interchanging x and y, the equation remains the same then the curve is symmetric about the line y=x.For ex: + = 3

(d)A curve is symmetric in the opposite quadrants if its equation remains the same where x and y replaced by –x and –y respectively.For ex: =

Symmetric about x-axis Symmetric about y-axis

Symmetric about a line y=x

2. (a) Curve through origin:




The curve passes through the origin, if the equation does not contain constant term.For ex: the curve =4ax passes through the origin.(b) Tangent at the origin:To know the nature of a multiple point it is necessary to find the tangent at that point.The equation of the tangent at the origin can be obtained by equating to zero, the lowest degree term in the equation of the curve.

3. The points of intersection with the axes:

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

For ex: + = 1 put y=0 we get = ±Thus, (a, 0) and (-a, 0) are the co-ordinates of point of intersection.

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

4. Regions in which the curve does not lie:

If the value of y is imaginary for certain value of x then the curve does not exist for such values.Example 1: = 4Answer: For negative value of x, if y is imaginary then there is no curve in second and third quadrants.Example 2: = (2 − ).Answer: (i) For y>2a, x is imaginary. There is no curve beyond y=2a.

(ii) For negative value of y, if x is imaginary then there is no curve in 3rd and 4th quadrants.

5. Asymptotes are the tangents to the curve at infinity:

(a)Asymptote parallel to the x-axis is obtained by equating to zero, the coefficient of the highest power of x.




For ex: − 4 + + 2 = 0 ( − 4) + + 2 = 0

The coefficient of the highest power of x . is − 4 = 0∴ − 4 = 0is the asymptote parallel to the x-axis.

(b) Asymptote parallel to the y-axis is obtained by equating to zero, the coefficient of highest power of y.

For ex: − 2 + + + 2 = 0 ( − 2) + + + 2 = 0

The coefficient of the highest power of . . is − 2.∴ − 2 = 0 is the asymptote parallel to y-axis.

(c) Oblique Asymptote: = +(I) Find ∅ ( ) by putting x=1 and y=m in highest degree (n) terms of

the equation of the curve.

(II) Solve ∅ ( ) = 0 for

(III) Find ∅ ( ) by putting x=1 and y=m in the next highest degree

(n-1) terms of the equation of the curve.

(IV)Find by the formula, = −∅ ( )∅′ ( ) , if the values of m are not

equal, then find by ∅′′ ( ) + ∅′ ( ) + ∅ ( ) = 0(V) Obtain the equation of asymptote by putting the values of m and c in = + .For ex: Find asymptote of + − 3 = 0Solution: Here, ∅ ( ) = 1 + and ∅ ( ) = −3Putting ∅ ( ) = 0 or + 1 = 0 ( + 1)( − + 1) = 0 = −1, = ±√




Only real value of m is −1.Now we find c from the equation

= −∅ ( )∅′ ( )

= −−33 = 1 = 1−1 = −1On putting = −1 and = −1 in = + , the equation of

asymptote is

= (−1) + (−1) + + 1 = 0

6. Tangent:

Put = 0 for the points where tangent is parallel to the x-axis.

For ex: + − 4 + 4 − 1 = 0 2 + 2 − 4 + 4 = 0 (2 + 4) = 4 − 2 =Now, = 04 − 2 = 0.

= 2Putting = 2 in (i), we get + 4 − 5 = 0

∴ = 1,−5The tangents are parallel to x-axis at the points (2,1) and (2,-5).

7. Table:




Prepare a table foe certain values of x and y and draw the curve passing through them.For Ex: = 4 + 4

X -1 0 1 2 3y 0 ±2 ±2√2 ±2√3 ±4

Note : Remember POSTER. Where,

P = point of intersection

O = Origin

S = Symmetry

T = Tangent

A = Asymptote

R = Region

3.1Trace the following curves:

Example 1: Trace the curve = ( − )Solution: we have,= ( − ) ________ (i)

1) Symmetry: Since the equation (i) contains only even power of y,∴it is symmetric about the x-axis.

It is not symmetric about y-axis since it does not contain even power of x.

2) Origin: Since constant term is absent in (i), it passes through origin.3) Intersection with x-axis:

Putting y=0 in (i), we get x=a.∴Curve cuts the x-axis at (a, 0).4) Tangent:The equation of the tangent at origin is obtained by equating

to zero the lowest degree term of the equation (i).


Unit-1

RAI UNIVERSITY, AHMEDABAD

= .

= = ±There are two tangents

Example 2: Trace ( +Solution: Here we have,

( + ) = ( −1) Origin: The equation of the given curve does not contain constant

term, therefore, the curve passes through origin.2) Symmetric about axes:

well as y, so the curve is symmetric about both the axes.3) Point of intersection wi

we get

4) Tangent at the origin: equating to zero the lowest degree term.

There are two tangents 5) Node: Origin is the node, since, there are two real and different

tangents at the origin.6) Region of absence of the curve: − , becomes negative, hence, the entire curve remains between = − =

CURVE TRACING

There are two tangents = ± at the origin to the given curve.

) = ( − )

) _____________(i)

The equation of the given curve does not contain constant term, therefore, the curve passes through origin.Symmetric about axes: The equation contains even powers of x as well as y, so the curve is symmetric about both the axes.Point of intersection with x-axis: On putting y=0 in the equation,

( − ) = 0, = ± , 0,0Tangent at the origin: Equation of the tangent is obtained by equating to zero the lowest degree term.− = 0 ⇒ = ±There are two tangents = and = − at the origin.

Origin is the node, since, there are two real and different tangents at the origin.Region of absence of the curve: For values of > and

becomes negative, hence, the entire curve remains between = .

CURVE TRACING

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at the origin to the given curve.

The equation of the given curve does not contain constant

The equation contains even powers of x as well as y, so the curve is symmetric about both the axes.

On putting y=0 in the equation,

Equation of the tangent is obtained by

Origin is the node, since, there are two real and different

and <becomes negative, hence, the entire curve remains between


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Example 3: Trace the curve

Solution: We have, = (1) Origin: Equation does not contain any constant term. Therefore, it passes

through origin.2) Symmetric about x-axis:

therefore, it is symmetric about x3) Tangent at the origin:

zero the lowest degree terms in the equation (i).

Equation of tangent:

2 = 0 ⟹ = 04) Cusp: As two tangents are coincident, therefore, origin is a cusp.5) Asymptote parallel to y

equating the coefficient of highest degree of y to zero.2 Equation of asymptote is

6) Region of absence of curve: > 2 < 0, therefore, the curve does not exist for< 0 > 2 .

CURVE TRACING

Trace the curve (2 − ) = (cissoid)

( ) _____________ (i)

Equation does not contain any constant term. Therefore, it passes

axis: Equation contains only even powers of y, therefore, it is symmetric about x-axis.Tangent at the origin: Equation of the tangent is obtained by equating to zero the lowest degree terms in the equation (i).2 − =

0, = 0is the double point.

As two tangents are coincident, therefore, origin is a cusp.Asymptote parallel to y-axis: Equation of asymptote is obtained by equating the coefficient of highest degree of y to zero.− = ⟹ (2 − ) =

Equation of asymptote is 2 − = 0 ⟹ = 2 .Region of absence of curve: becomes negative on putting

, therefore, the curve does not exist for

CURVE TRACING

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_____________ (i)

Equation does not contain any constant term. Therefore, it passes

Equation contains only even powers of y,

Equation of the tangent is obtained by equating to

As two tangents are coincident, therefore, origin is a cusp.Equation of asymptote is obtained by

becomes negative on putting


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Example 4: Trace the curve

Solution: we have, =1) Symmetry:

(a) The curve is symmetric about y(b) Not symmetric about x

2) Origin: The curve passes through the origin contain any constant term.

3) Region of absence of the curve: becomes negative but left hand side becomes positive hence, the curve does not exist when y=2.

4) Tangent at the origin:

2 ⟹ = 25) Intercept on y-axis: On putting x=0 in the equation, we get

CURVE TRACING

Trace the curve = (2 − )(2 − ) _________(i)

(a) The curve is symmetric about y-axis. Since all the powers of x are even.(b) Not symmetric about x-axis. Since all the powers of y are not even.

The curve passes through the origin since the equation does not contain any constant term.Region of absence of the curve: If y is greater than 2a the right hand side becomes negative but left hand side becomes positive hence, the curve does

Tangent at the origin: On putting the lowest degree term to zero

⟹ = ±On putting x=0 in the equation, we get⟹ 0 = (2 − )

⟹ 2 − = 0⟹ = 2

CURVE TRACING

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axis. Since all the powers of x are even.axis. Since all the powers of y are not even.

since the equation does not

If y is greater than 2a the right hand side becomes negative but left hand side becomes positive hence, the curve does

ing the lowest degree term to zero =


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4.1 SOME IMPORTANT CURVES

CURVE TRACING

SOME IMPORTANT CURVES

CURVE TRACING

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Unit-1


CURVE TRACINGCURVE TRACING

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5.1 EXERCISE:

Trace the following curves.1) = 4 (2 − )2) (4 − ) =3) + = 54) 3 = ( − )

6.1 REFERENCE BOOK:

1) Introduction to Engineering MathematicsBy H. K. DASS.& Dr. RAMA VERMA

2) WWW.brightclasses.in3) www.sakshieducation.com/.../M1...EnvelopesCurveTracing.pdf4) Higher Engineering Mathematics

By B.V.RAMANA


B.Tech-II_Unit-I

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