Higher Engineering Mathematics, Sixth Edition

Functions and their curves 199

x^2 +y^2 = 4 .Comparing this equation with

x^2 +y^2 =a^2 shows thatx^2 +y^2 =4 is the equa-

tion of acirclehaving centre at the origin (0, 0)

and of radius 2 units.

(b) Transposing

y^2

8

= 2 x gives y= 4

√

x. Thus

y^2

8

= 2 xis the equation of aparabolahaving its

axis of symmetry coinciding with thex-axis and

its vertex at the origin of a rectangular co-ordinate

system.

(c) y= 6

(

1 −

x^2

16

) 1 / 2

can be transposed to

y

6

=

(

1 −

x^2

16

) 1 / 2

and squaring both sides gives

y^2

36

= 1 −

x^2

16

,i.e.

x^2

16

+

y^2

36

=1.

This is the equation of anellipse, centre at the ori-

gin of a rectangular co-ordinate system, the major

axis coinciding with they-axis and being 2

√

36,

i.e. 12 units long. The minor axis coincides with

thex-axis and is 2

√

16, i.e. 8 units long.

Problem 17. Describe the shape of the curves

represented by the following equations:

(a)

x

5

=

√[

1 +

(y

2

) 2 ]

(b)

y

4

=

15

2 x

(a) Since

x

5

=

√[

1 +

(y

2

) 2 ]

x^2

25

= 1 +

(y

2

) 2

i.e.

x^2

25

−

y^2

4

= 1

This is ahyperbolawhich is symmetrical about

both thex-andy-axes, the vertices being 2

√

25,

i.e. 10 units apart.

(With reference to Section 18.1 (vii),ais equal

to±5)

(b) The equation

y

4

=

15

2 x

is of the formy=

a

x

,a=

60

2

=30.

This represents arectangular hyperbola,sym-

metrical about both thex-andy-axis, and lying

entirely in the first and third quadrants, similar in

shape to the curves shown in Fig. 18.9.

Now try the following exercise

Exercise 81 Further problems on curve

sketching

1. Sketch the graphs of (a) y= 3 x^2 + 9 x+

7

4

(b)y=− 5 x^2 + 20 x+50.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

(a) Parabola with minimum

value at

(

−^32 ,− 5

)

and

passing through

(

0 , (^134)
)
.
(b) Parabola with maximum
value at( 2 , 70 )and passing
through( 0 , 50 ).
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
In Problems 2 to 8, sketch the curves depicting the
equations given.

2. x= 4

√[

1 −

(y

4

) 2 ]

[circle, centre (0, 0), radius 4 units]

3.

√

x=

y

9

[

parabola,symmetrical about

x-axis,vertex at( 0 , 0 )

]

4. y^2 =

x^2 − 16

4

⎡

⎢

⎢

⎣

hyperbola,symmetrical about

x-andy-axes,distance

between vertices 8 units along

x-axis

⎤

⎥

⎥

⎦

5.

y^2

5

= 5 −

x^2

2

⎡

⎣

ellipse,centre( 0 , 0 ),major axis

10 units alongy-axis,minor axis

2

√

10 units alongx-axis

⎤

⎦

6. x= 3

√

1 +y^2

⎡

⎢

⎢

⎣

hyperbola,symmetrical about

x-andy-axes,distance

between vertices 6 units along

x-axis

⎤

⎥

⎥

⎦

7. x^2 y^2 = (^9) [
   rectangular hyperbola,lying in
   first and third quadrants only
   ]