Higher Engineering Mathematics

FUNCTIONS AND THEIR CURVES 211

C

Figure 19.37

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 2brepre-

sent the length of the axes of the ellipse. Thus

x^2

32

+

y^2

22

=1 represents an ellipse, having its

axes coinciding with thex- andy-axes of a rect-

angular 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. 19.38(a).

4

6

x

y

(a) 4x^2 = 36 − 9 y^2

x

y

(b) 3y^2 + 15 = 5 x^2

2 √ 3

Figure 19.38

(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

symmetrical about both thex- andy-axes, the

distance between the vertices being given by 2a.

Thus a sketch of

x^2

3

−

y^2

5

=1isasshown

in Fig. 19.38(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