Higher Engineering Mathematics, Sixth Edition

198 Higher Engineering Mathematics

x

3

y

1

14

24

x

y

x

y

(a) x 5! (9 2 y^2 )

(b) y^2516 x

(c) xy 55

Figure 18.37

Problem 15. Sketch the curves depicting the

following equations:

(a) 4x^2 = 36 − 9 y^2 (b) 3y^2 + 15 = 5 x^2

(a) By dividing throughout by 36 and transposing,

the equation 4x^2 = 36 − 9 y^2 can be written as

x^2

9

+

y^2

4

=1. The equation of an ellipse is of the

form

x^2

a^2

+

y^2

b^2

=1, where 2aand 2brepresent the

length of theaxes of the ellipse.Thus

x^2

32

+

y^2

22

= 1

represents an ellipse, having its axes coinciding

withthex-andy-axes of a rectangular co-ordinate

system, the major axis being 2(3), i.e. 6 units long

and the minor axis 2(2), i.e. 4 units long, as shown

in Fig. 18.38(a).

4

6

x

y

x

y

(a) 4 x^253629 y^2

(b) 3 y^211555 x^2

2 3Œ„

Figure 18.38

(b) Dividing 3y^2 + 15 = 5 x^2 throughout by 15 and

transposing gives

x^2

3

−

y^2

5

=1. The equation

x^2

a^2

−

y^2

b^2

=1 represents a hyperbola which is sym-

metrical about both thex-andy-axes, the distance

between the vertices being given by 2a.

Thus a sketch of

x^2

3

−

y^2

5

=1isasshownin

Fig. 18.38(b), having a distance of 2

√

3 between

its vertices.

Problem 16. Describe the shape of the curves

represented by the following equations:

(a)x= 2

√[

1 −

(y

2

) 2 ]

(b)

y^2

8

= 2 x

(c)y= 6

(

1 −

x^2

16

) 1 / 2

(a) Squaring the equation givesx^2 = 4

[

1 −

(y

2

) 2 ]

and transposing givesx^2 = 4 −y^2 ,i.e.