Higher Engineering Mathematics, Sixth Edition

180 Higher Engineering Mathematics

x^2 y^2

a^2 b^2

1 51

y C b O a D

x

A B

Figure 18.7

In the above equation, ‘a’ is the semi-major axis and

‘b’ is the semi-minor axis.

(Note that ifb=a, the equation becomes

x^2

a^2

+

y^2

a^2

=1,

i.e.x^2 +y^2 =a^2 , which is a circle of radiusa).

(vii) Hyperbola

The equation of a hyperbola is

x^2

a^2

−

y^2

b^2

= 1

and the general shape is shown in Fig. 18.8. The

curve is seen to be symmetrical about both the

x-andy-axes. The distanceABin Fig. 18.8 is given

by 2a.

AB

y

O x

x^2 y^2

a^2 b^2

2 51

Figure 18.8

(viii) Rectangular Hyperbola

The equation of a rectangular hyperbola isxy=cor

y=

c

x

and the general shape is shown in Fig. 18.9.

(ix) Logarithmic Function(see Chapter 3, page 26)

y=lnxandy=lgxarebothofthegeneral shapeshown

in Fig. 18.10.

(x) Exponential Functions(see Chapter 4, page 30)

y=exis of the general shape shown in Fig. 18.11.

23 22 21 12 3

21

22

23

1

2

3

0

y 5

y

c

x

x

Figure 18.9

01

y 5 log x

y

x

Figure 18.10

(xi) Polar Curves

The equation of a polar curve is of the formr=f(θ ).

An example of a polar curve,r=asinθ,isshownin

Fig. 18.12.