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CHAPTER 4
ASYMPTOTES
4.1 Introduction:
An asymptote is a line 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 4 ๐ฅ๐ฅ 2 + 9๐ฆ๐ฆ 2 = ๐ฅ๐ฅ 2 ๐ฆ๐ฆ 2 Solution: The equation of the given curve is 4 ๐ฅ๐ฅ 2 + 9๐ฆ๐ฆ 2 โ ๐ฅ๐ฅ 2 ๐ฆ๐ฆ 2 = 0 Equating it to zero, coefficient of ๐ฅ๐ฅ 2 (which is highest power of x ) we get, 4 โ ๐ฆ๐ฆ 2 = 0 โน ๐ฆ๐ฆ = ยฑ โด ๐ฆ๐ฆ = 2, ๐ฆ๐ฆ = โ 2 are the two asymptotes parallel to x -axis Equating to zero the coefficient of highest power of y , we get
โด ๐ฅ๐ฅ = 3, ๐ฅ๐ฅ = โ 3 are the asymptotes parallel to y โ axis
4.3 Method of finding oblique asymptote:
Let the asymptote be ๐ฆ๐ฆ = ๐๐๐ฅ๐ฅ + ๐๐ Let the equation of the curve be ๐๐๐๐(๐ฅ๐ฅ, ๐ฆ๐ฆ) + ๐๐ (^) ๐๐โ1(๐ฅ๐ฅ, ๐ฆ๐ฆ) +โฆโฆโฆ.+๐๐ 1 (๐ฅ๐ฅ, ๐ฆ๐ฆ) + ๐๐ = 0 โ (1) where ๐๐๐๐ (๐ฅ๐ฅ, ๐ฆ๐ฆ)^ denotes the term of highest degree of the curve. Step-1 Put x = 1, y = m in ๐๐๐๐(x, y), ฯnโ1 (x, y), โฆโฆ,๐๐ 1 (๐ฅ๐ฅ, ๐ฆ๐ฆ) Step-2 Find all the real roots of ๐๐๐๐(m) = 0 Step-3 If m is a non-repeated root, then corresponding value of c is given by c ๐๐๐๐โฒ^ (m) + ๐๐ (^) nโ1 (m) = 0, (๐๐๐๐โฒ^ (m) โ 0) If ๐๐๐๐โฒ^ (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 ๐๐ 2 2! ๐๐๐๐
โฒโฒ (^) (m) + ๐๐ 1! ๐๐^ ๐๐โ
โฒ (^) (m) + ๐๐ (^) ๐๐โ2 (m) = 0 (๐๐๐๐โฒโฒ (^) (m) (^) โ 0)
Step-5 The asymptote of the curve is y = m x + c
Example2 Find all the asymptotes of the curve ๐ฅ๐ฅ 3 โ ๐ฅ๐ฅ 2 ๐ฆ๐ฆ โ ๐ฅ๐ฅ๐ฆ๐ฆ 2 + ๐ฆ๐ฆ 3 + 2๐ฅ๐ฅ 2 โ 4๐ฆ๐ฆ 2 + 2๐ฅ๐ฅ๐ฆ๐ฆ + ๐ฅ๐ฅ + ๐ฆ๐ฆ + 1 = 0 Solution: In the curve the highest degree term of x is ๐ฅ๐ฅ 3 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 โ axis. Also the coefficient of highest degree term in y is 1, thus the curve has no asymptote parallel to y-axis. Now finding oblique asymptote Here ๐๐ 3 (x, y) = ๐ฅ๐ฅ 3 โ ๐ฅ๐ฅ 2 ๐ฆ๐ฆ + ๐ฆ๐ฆ 2 โ ๐ฅ๐ฅ๐ฆ๐ฆ 2 ๐๐ 2 (x, y) = 2๐ฅ๐ฅ 2 โ 4๐ฆ๐ฆ 2 + 2๐ฅ๐ฅ๐ฆ๐ฆ , ๐๐ 1 (x, y) = ๐ฅ๐ฅ + ๐ฆ๐ฆ Let the asymptote be given by ๐ฆ๐ฆ = ๐๐๐ฅ๐ฅ + ๐๐ Step-1 Putting x = 1 & y = m in ๐๐ 3 (x, y), ๐๐ 2 (x, y), ๐๐ 1 (x, y) ๐๐ 3 (m) = ๐๐ 3 โ ๐๐ 2 โ ๐๐ + 1 ๐๐ 2 (m) = 2 โ 4๐๐ 2 + 2๐๐ and ๐๐ 1 (m) = 1 + m Step-2 The values of m are obtained by solving ๐๐ 3 (m) = 0
โด ๐๐ 3 โ ๐๐ 2 โ ๐๐ + 1 = 0 โ (๐๐ 2 โ 1) (๐๐ โ 1) = 0 โ m = 1, 1, โ 1 Here m = โ1 is a non repeated root & m = 1 is a repeated root. Step-3: For m = โ 1 (non-repeated root),the corresponding value of c is given by c๐๐ 3 โฒ^ (m) + ๐๐ 2 (m) = 0 Now ๐๐ 3 (m) = ๐๐ 3 โ ๐๐ 2 โ ๐๐ + 1 โ ๐๐ 3 โฒ^ = 3 ๐๐ 2 โ 2๐๐ โ 1 โด ๐๐(3๐๐ 2 โ 2๐๐ โ 1) + (2 + 2๐๐ โ 4๐๐ 2 ) = 0 โ ๐๐๏ฟฝ 3 ๐๐ 2 โ 2๐๐ โ 1๏ฟฝ = 4๐๐ 2 โ 2๐๐ โ 2 โ ๐๐ =
4๐๐ 2 โ 2๐๐ โ 2 (3๐๐ 2 โ 2๐๐ โ 1 ) =^
4 4 = 1
๐ถ๐ถ 2 2! ๐๐^3
โฒโฒ(m) + ๐๐๐๐ 2
โฒโฒ (^) (m) + ๐๐ 1 (m) = 0 ๐ถ๐ถ 2 2 (6๐๐^ + 4) +^ ๐๐(0) โ^ ๐๐^ = 0 โ (3๐๐ + 2) ๐๐ 2 โ ๐๐ = 0 Putting m = โ 1 in this equation, we get
- ๐๐ 2 + 1 = 0 โ ๐๐ 2 = 1 โ C = ยฑ 1 โด Asymptote corresponding to m = โ 1, c = 1 is ๐ฆ๐ฆ = โ ๐ฅ๐ฅ + 1 โ ๐ฆ๐ฆ + ๐ฅ๐ฅ โ 1 = 0 And asymptote corresponding to m = โ 1, c = โ1 is y = โx โ 1 โ ๐ฆ๐ฆ + ๐ฅ๐ฅ โ 1 = 0 Thus the asymptotes of the given curve are ๐ฆ๐ฆ = 0, ๐ฆ๐ฆ + ๐ฅ๐ฅ โ 1 = 0 & ๐ฆ๐ฆ + ๐ฅ๐ฅ + 1 = 0
Note: If m is a repeated root occurring thrice then the values of c are given by ๐๐ 3 3! ๐๐๐๐
โฒโฒโฒ (^) (m) + ๐๐^2 2! ๐๐^ ๐๐โ
โฒโฒ (^) (m) + ๐๐ 1! ๐๐^ ๐๐โ
โฒ (^) (m) + ๐๐n-3 (m) = 0
Example 4 Find the asymptotes of the curve ๐ฆ๐ฆ 4 โ 2๐ฅ๐ฅ๐ฆ๐ฆ 3 + 2๐ฅ๐ฅ 3 ๐ฆ๐ฆ โ ๐ฅ๐ฅ 4 โ 3๐ฅ๐ฅ 3 + 3๐ฅ๐ฅ 2 ๐ฆ๐ฆ + 3๐ฅ๐ฅ๐ฆ๐ฆ 2 โ 3๐ฆ๐ฆ 3 โ 2๐ฅ๐ฅ 2 + 2๐ฆ๐ฆ 2 โ 1 = 0 Solution: In this case, vertical and horizontal asymptotes do not exist. To find oblique asymptotes:
Here ๐๐ 4 (x, y) = ๐ฆ๐ฆ 4 โ 2๐ฅ๐ฅ๐ฆ๐ฆ 3 + 2๐ฅ๐ฅ 3 ๐ฆ๐ฆ โ ๐ฅ๐ฅ 4
๐๐ 3 (x, y) = โ 3๐ฅ๐ฅ 3 + 3๐ฅ๐ฅ 2 ๐ฆ๐ฆ + 3๐ฅ๐ฅ๐ฆ๐ฆ 2 โ 3๐ฆ๐ฆ 3 โ (1)
๐๐ 2 (x, y) = โ2x^2 + 2x^2
๐๐ 1 (x, y) = 0
Step - 1 Putting x = 1, y = m in (1), we get
๐๐ 4 (m) = ๐๐ 4 โ 2๐๐ 3 + 2๐๐ โ 1 ๐๐ 3 (m) = โ 3 + 3๐๐ + 3๐๐ 2 โ 3๐๐ 3 ๐๐ 2 (m) = โ 2 + 2๐๐ 2 ๐๐ 1 (m) = 0 Step -2 The values of m are obtained by solving ๐๐ 4 (m) = 0
โ ๐๐ 4 โ 2๐๐ 3 + 2๐๐ โ 1 = 0 โ (๐๐โ 1)^3 (๐๐ + 1) = 0 โ m = 1, 1, 1, โ Step -3 For m = โ1 (non โ repeated root), the value of c is given by
c= โ (^) ๐๐๐๐^3 (m) 4 โฒ^ (m)^
= โ^3 (โm
(^3) +m (^2) +mโ 1 ) 3 ๏ฟฝmโ1๏ฟฝ^2 (m+1)+ ๏ฟฝmโ1๏ฟฝ.^3 = 0 (for m= -1) โด y = โ x is the asymptote corresponding to m = โ 1 Step -4 For m = 1 (repeated root), we have
๐๐ 3 3! ๐๐^4
โฒโฒ (^) (m) + ๐๐^2 2! ๐๐^3
โฒโฒ (^) (m) + c ๐๐ 2
โฒ (^) (m) + ๐๐ 1 (m) = 0 โ
๐๐ 3 6 (24^ ๐๐^ โ 12) +^
๐๐ 2 2 ๏ฟฝโ 18๐๐^ + 6๏ฟฝ^ +^ ๐๐. 4๐๐^ + 0 = 0 โ ๐๐ [ (4๐๐ โ 2) ๐๐ 2 + (โ 9๐๐ + 3) ๐๐ + 4 ๐๐ ] = 0 For m = 1, we get c = 0 and 2 ๐๐ 2 โ 6๐๐ + 4 = 0 โ (๐๐ โ 1) (๐๐ โ 2) = 0 โด Values of c are 0, 1, 2 โด Asymptotes are ๐ฆ๐ฆ = ๐ฅ๐ฅ, ๐ฆ๐ฆ = ๐ฅ๐ฅ + 1 , ๐ฆ๐ฆ = ๐ฅ๐ฅ + 2 and ๐ฆ๐ฆ = โ ๐ฅ๐ฅ
Example 5 Find the asymptotes of the curve
(๐ฅ๐ฅ 2 โ ๐ฆ๐ฆ 2 ) (๐ฅ๐ฅ + 2๐ฆ๐ฆ) + 5 (๐ฅ๐ฅ 2 + ๐ฆ๐ฆ 2 ) + ๐ฅ๐ฅ + ๐ฆ๐ฆ = 0 Solution: The curve has neither horizontal asymptote nor vertical asymptote.
Here ๐๐ 3 (x, y) = (๐ฅ๐ฅ 2 โ ๐ฆ๐ฆ 2 ) (๐ฅ๐ฅ + 2๐ฆ๐ฆ)
๐๐ 2 (๐ฅ๐ฅ, ๐ฆ๐ฆ) = 5 (๐ฅ๐ฅ 2 + ๐ฆ๐ฆ 2 ) โ(1)
๐๐ 1 (x, y) = x + y
Step -1 Putting x = 1 and y = m in (1), we get
๐๐ 3 (m) = (1 โ ๐๐ 2 ) (1 + 2๐๐) ๐๐ 2 (m) = 5 (1 + ๐๐ 2 ) ๐๐ 1 (m) = 1 + m Step -2 The values of m are obtained by ๐๐ 3 (m) = 0 โ (1 โ ๐๐ 2 ) (1 + 2๐๐) = 0 โ m = ยฑ 1,
โ 2 Step -3 For m = ยฑ1,
โ 2 we have c = โ
๐๐ 2 (m) ๐๐ 3 โฒ^ (m)
(Since all are non repeated roots)
Now m = 1 gives ๐๐ =
5 3 โ^ y =^
๐ฅ๐ฅ+ 3 is the asymptote m = โ 1 gives ๐๐ =
- 5 3 โ^ y = โ^ ๐ฅ๐ฅ โ^
5 3 is the asymptote m = โ
1 2 gives c = โ^
25 6 โ^ y = โ^
๐ฅ๐ฅ 2 โ^
25 6 is the asymptote โด All the asymptotes are 3 ๐ฆ๐ฆ โ 3 ๐ฆ๐ฆ + 5 = 0, 3 ๐ฅ๐ฅ + 3๐ฆ๐ฆ + 5 = 0 and 3 ๐ฅ๐ฅ + 6๐ฆ๐ฆ + 25 = 0
