Functions and their curves 199
x
2
+ y
2
= 4. Comparing this equation with
x 2 + y 2 = a 2 shows that x 2 + y 2 = 4 is the equation of a circle having centre at the origin (0, 0)
and of radius 2 units.
(b) Transposing
y 2
8
= 2x gives y = 4
√
x. Thus
y 2
8
= 2x is the equation of a parabola having its
axis of symmetry coinciding with the x-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 an ellipse, centre at the origin of a rectangular co-ordinate system, the major
axis coinciding with the y-axis and being 2
√
36,
i.e. 12 units long. The minor axis coincides 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
2x
(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 a hyperbola which is symmetrical about
both the x- and y-axes, the vertices being 2
√
25,
i.e. 10 units apart.
(With reference to Section 18.1 (vii), a is equal
to ±5)
(b) The equation
y
4
=
15
2x
is of the form y =
a
x
, a =
60
2
= 30.
This represents a rectangular hyperbola, symmetrical about both the x- and y-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 = 3x 2 + 9x +
7
4
(b) y =−5x 2 + 20x + 50.
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
(a) Parabola with minimum
value at
−
3
2 , −5
and
passing through
0, 1
3
4
.
(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- and y-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 along y-axis, minor axis
2
√
10 units along x-axis
⎤
⎦
6. x = 3
1 + y 2
⎡
⎢
⎢
⎣
hyperbola, symmetrical about
x- and y-axes, distance
between vertices 6 units along
x-axis
⎤
⎥
⎥
⎦
7. x 2 y 2 = 9
rectangular hyperbola, lying in
first and third quadrants only
x
2
+ y
2
= 4. Comparing this equation with
x 2 + y 2 = a 2 shows that x 2 + y 2 = 4 is the equation of a circle having centre at the origin (0, 0)
and of radius 2 units.
(b) Transposing
y 2
8
= 2x gives y = 4
√
x. Thus
y 2
8
= 2x is the equation of a parabola having its
axis of symmetry coinciding with the x-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 an ellipse, centre at the origin of a rectangular co-ordinate system, the major
axis coinciding with the y-axis and being 2
√
36,
i.e. 12 units long. The minor axis coincides 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
2x
(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 a hyperbola which is symmetrical about
both the x- and y-axes, the vertices being 2
√
25,
i.e. 10 units apart.
(With reference to Section 18.1 (vii), a is equal
to ±5)
(b) The equation
y
4
=
15
2x
is of the form y =
a
x
, a =
60
2
= 30.
This represents a rectangular hyperbola, symmetrical about both the x- and y-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 = 3x 2 + 9x +
7
4
(b) y =−5x 2 + 20x + 50.
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
(a) Parabola with minimum
value at
−
3
2 , −5
and
passing through
0, 1
3
4
.
(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- and y-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 along y-axis, minor axis
2
√
10 units along x-axis
⎤
⎦
6. x = 3
1 + y 2
⎡
⎢
⎢
⎣
hyperbola, symmetrical about
x- and y-axes, distance
between vertices 6 units along
x-axis
⎤
⎥
⎥
⎦
7. x 2 y 2 = 9
rectangular hyperbola, lying in
first and third quadrants only