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.