[PDF] ASYMPTOTES - Engineering Mathematics

ASYMPTOTES - Engineering Mathematics

An asymptote is a line that approachescloser to a given curve as one or both of or x y coordinates tend to infinitybut never intersects or crosses the curve There are two types of asymptotes viz Rectangular asymptotes and Oblique asymptotes Rectangular Asymptote: If an asymptote is parallel to or to x-axis y -

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CHAPTER 4

ASYMPTOTES

4.1 Introduction:

An asymptote is a li

ne that approaches closer to a given curve as one or both of x or y coordinates tend to infinity but never intersects or crosses the curve. There are two types of asymptotes viz. Rectangular asymptotes and Oblique asymptotes Rectangular Asymptote:

If an asymptote is parallel to

x axis or to y- axis, then it is called rectangular asymptote.

An asymptote parallel to

x axis is called horizontal asymptote and the asymptote parallel to y-axis is called vertical asymptote.

Oblique

Asymptote:

If an asymptote is neither parallel to x-axis

nor to y axis then it is called an oblique asymptote. An oblique asymptote occurs when the degree of polynomial in the numerator is greater than that of polynomial in the denominator. To find the oblique asymptote, numerator must be divided by the denominator by using either long division or synthetic division.

4.2 Method of finding rectangular asymptote: To find an asymptote parallel to x-axis equate to zero the coefficient of highest

power of x in the equation of the curve. To find an asymptote parallel to y-axis equate to zero the coefficient of highest power of y in the equation of the curve.

Example1

Find the asymptotes parallel to coordinate axes of the curve + 9ݕ

Solution: The equation of the given curve is 4

+ 9ݕ =0

Equating it to zero, coefficient of ݔ

(which is highest power of x) we get,

4െݕ

=0 ฺ

ݕ= ±2

Equating to zero the coefficient of highest power of y , we get

9െݔ

=0 ฺݔ= ±3

4.3 Method of finding oblique asymptote:

Let the asymptote be ݕ=݉ݔ+ܿ

Let the equation of the curve be

(ݔ,ݕ)+ ݇=0 െ(1) where (ݔ,ݕ) denotes the term of highest degree of the curve. (x,y),Ԅ (m)=0 Step-3 If m is a non-repeated root, then corresponding value of c is given by (m)്0) (m) = 0 then there is no asymptote to the curve corresponding to this value of m. Step-4 If m is a repeated root occurring twice, then the two values of c are given by (m) + (m)്0)

Step-5 The asymptote of the curve is y = m x + c

Example2

Find all the asymptotes of the curve

+ 2ݔ - 4ݕ + 2ݔݕ + ݔ + ݕ + 1 = 0 Solution: In the curve the highest degree term of x is ݔ and its coefficient is 1.Equating it to 0 we get

1= 0 which is absurd, thus the curve has no asymptote parallel to x - axis.

Also the coefficient of highest degree term in y is 1, thus the curve has no a symptote parallel to y-axis.

Now finding obliq

ue asymptote H ere (x,y) = ݔ (x,y) =2ݔ - 4ݕ (x,y) = ݔ + ݕ Let the asymptote be given by ݕ = ݉ݔ + ܿ (x,y) (m) = ݉ - ݉ + 1 (m) = 2 - 4݉

3 (m) = 0

- ݉ + 1 = 0 ֜ -1) (݉െ1) = 0 Here m = -1 is a non repeated root & m = 1 is a repeated root. Step-3: For m = - 1 (non-repeated root),the corresponding value of c is given by (m) = 0 (m) =݉ - ݉ + 1 = 3݉ - 2݉ -1 - 2݉ - 1) + (2 + 2݉ - 4݉ ) = 0 - 2݉ - 1൯= 4݉ - 2݉ - 2 ܿ ֜ =1

Thus for ݉= - 1, c= 1

Now the asymp

tote is ݕ=݉ݔ+ܿ Step-4 For m = 1 (repeated root), the value of c is given by (m) + (m) = 0........(2) (m) = 3݉ - 2݉ - 1 (m)=6mെ2 (m)=2 + 2݉ - 4݉ (m)=2െ8m

1 (m) = 1 + m

Putting these values in (2) we get

൫6݉-2൯+ ܿ (4) + ܿ (-6) + 2 = 0 ܿ ֜ -3ܿ 6 6

For m = 1, c=

For m = 1,

c 6 ݔ + ݕ = 1,2(ݕ-ݔ) = 3 + ξ5 ,2(ݕ-ݔ) = 3 - ξ5

Example

3 Find the asymptotes of the curve ݕ

ݕ + 2ݔݕ

- ݕ + 1 = 0

Solution: The highest degree term of y is y

3 whose coefficient is 1. Equating it to 0 we get 1 = 0 which is not possible ׵ of highest degree term in x is y thus y = 0 is the asymptote parallel to x - axis.

To find oblique asymptote:

(x,y) = ݕ

ݕ + 2ݔݕ

(x,y)= - y, Step -1 Putting x = 1, y = m in the above functions, we get + ݉ + 2݉ (݉)=3݉ + 4݉ + 1

3 (m) = 0

i.e. ݉ + 2݉ + ݉ = 0 +2݉+1) = 0 ֜ Step - 3 For m = 0 (non-repeating value), the value of c is given by c = 0

Step - 4 For m = - 1, - 1 (repeated root)

(m) = 0 (6݉+4) + ܿ ֜ (3݉ + 2) ܿ - ݉ = 0

Putting m =

1 in this equation, we get

+ 1 = 0 ܿ ֜ = 1 ֜

ݕ = - ݔ + 1 ֜

And asymptote corresponding to m = - 1, c = -1 is

y = x - 1 ֜

Thus the asymptote

s of the given curve are ݕ = 0,ݕ + ݔ - 1 = 0 & ݕ + ݔ + 1 = 0 Note: If m is a repeated root occurring thrice then the values of c are given by (m)+ (m) +

Example

4 Find the asymptotes of the curve

-2ݔݕ + 2ݔ - 3ݔ + 3ݔ

ݕ + 3ݔݕ

- 3ݕ - 2ݔ + 2ݕ -1 = 0 Solution: In this case, vertical and horizontal asymptotes do not exist.

To find oblique asymptotes

(x,y) =ݕ - 2ݔݕ + 2ݔ (x,y) = - 3ݔ + 3ݔ

ݕ + 3ݔݕ

- 3ݕ (1) (x,y)=െ2x +2x (x,y) = 0

Step -1 Putting x = 1, y = m in (1), we get

(m) =݉ - 2݉ + 2݉ - 1 (m) = - 3 + 3݉ + 3݉ - 3݉ (m) = - 2 + 2݉ (m) = 0 - 2݉ + 2݉ - 1 = 0 (݉ + 1) = 0 Step -3 For m = -1 (non - repeated root), the value of c is given by c= - = 0 (for m= 1)

Step -4 For m = 1 (repeated root), we have

(m) + (m) = 0 (24 ݉ - 12) + k-18݉ + 6൯+ ܿ

ܿ֜ [ (4݉ - 2) ܿ

+ (- 9݉ + 3) ܿ

For m = 1, we get c = 0 and 2ܿ

- 6ܿ

֜ (ܿ - 1) (ܿ

Example 5

Find the asymptotes of the curve

) (ݔ + 2ݕ) +5 (ݔ ) + ݔ + ݕ = 0 Solution: The curve has neither horizontal asymptote nor vertical asymptote. (x,y) = (ݔ ) (ݔ + 2ݕ) (ݔ,ݕ) = 5 (ݔ ) -(1) (x,y) = x + y

Step -1 Putting x = 1 and y = m in (1), we get

(m) = (1 - ݉ ) (1 + 2݉) (m) = 5 (1 + ݉ (m) = 1 + m ) (1 + 2݉) = 0

Step -3 For m = ±1,

we have c = - (Since all are non repeated roots)

Now m = 1 gives ܿ

is the asymptote m =

1 gives ܿ

is the asymptote m = - gives c = - is the asymptote

3ݕെ 3 ݕ + 5 = 0, 3ݔ + 3ݕ + 5 = 0 and 3ݔ + 6ݕ + 25 = 0

Exercise 4A

1. Find the asymptotes of the following curves:

(i) = 1

Ans. x =

± b, y = ± a

(ii) + 3ݔ

ݕ - 4ݕ

- ݔ + ݕ + 3=0 Ans. ݕ = ݔ,ݔ + 2ݕ = 1 and ݔ + 2ݕ + 1 = 0 i ii) ݕ (ݔ - ݕ)

Ans. y = 0,y = x +ξ2 and y = x -ξ2

(iv) ݔ + 3ݔ = 0

Ans. y = 0 and y = -x

(v) ݕ - 2ݔݕ

ݕ + 2ݔ

+ 2ݔ - 3ݔݕ + ݔ - 2ݕ + 1 = 0 Ans. 2 (ݔ - ݕ) = 1,6 (ݔ + ݕ) + 5 = 0 and 6ݔ - 3 ݕ + 4 = 0 (vi) - 4ݔ + ݔ = 0 Ans. y = ݔ + 1,ݕ = ݔ - 1,ݕ = - ݔ + 1,ݕ = - ݔ - 1 (vii) ݔ + 4ݔ

ݕ + 5ݔݕ

+ 2ݕ + 2ݔ + 4ݔݕ + 2ݕ - ݔ - 9ݕ + 1 = 0 Ans. ݔ+2ݕ+2 = 0, ݔ + ݕ ± 2ξ2 = 0 (viii) (ݔ - ݕ) (ݔ - 2ݕ) (ݔ - 3ݕ) - 2ܽ ) - (ݔ - 2ݕ) (ݔ + ݕ) = 0 Ans. 2ݕ = ݔ + 14ܽ,3ݕ = ݔ - 13ܽ,ݕ = ݔ - ܽ,ݕ = ݔ - 2ܽquotesdbs_dbs11.pdfusesText_17