Higher Engineering Mathematics

212 GRAPHS

(a) Squaring the equation givesx^2 = 4

[

1 −

(y

2

) 2 ]

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

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

origin 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 coin-

cides with the x-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 19.1 (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. 19.9.

Now try the following exercise.

Exercise 89 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, 1^34

)

.

(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
⎤
⎥
⎦