trigonometry

142 GEOMETRY AND TRIGONOMETRY
Problem 13. Sketch the circle given by the
equation: x 2 + y 2 − 4x + 6y − 3 = 0.
The equation of a circle, centre (a, b), radius r is
given by:
(x − a) 2 + (y − b) 2 = r 2
The general equation of a circle is
x 2 + y 2 + 2ex + 2fy + c = 0.
From above a =− 2e
2 , b =−2f 2 and
r = √ (a 2 + b 2 − c).
Hence if x 2 + y 2 − 4x + 6y − 3 = 0
then
and
a =− ( ) (
−4
2
= 2, b =−
62
)
=−3
r = √ [(2) 2 + (−3) 2 − (−3)]
= √ 16 = 4
Thus the circle has centre (2, −3) and radius 4,as
shown in Fig. 14.11.
Alternatively, x 2 + y 2 − 4x + 6y − 3 = 0 may be
rearranged as:
(x − 2) 2 + (y + 3) 2 − 3 − 13 = 0
i.e. (x − 2) 2 + (y + 3) 2 = 4 2
which represents a circle, centre (2, −3) and
radius 4, as stated above.
y
4
Now try the following exercise.
Exercise 66 Further problems on the equation
of a circle
1. Determine the radius and the co-ordinates of
the centre of the circle given by the equation
x 2 + y 2 + 6x − 2y − 26 = 0.
[6, (−3, 1)]
2. Sketch the circle given by the equation
x 2 + y 2 − 6x + 4y − 3 = 0.
[Centre at (3, −2), radius 4]
3. Sketch the curve x 2 + (y − 1) 2 − 25 = 0.
[Circle, centre (0, 1), radius 5]
4. Sketch the curve x = 6√ [1
− (y/6)
2 ] .
[Circle, centre (0, 0), radius 6]
14.6 Linear and angular velocity
Linear velocity
Linear velocity v is defined as the rate of change
of linear displacement s with respect to time t. For
motion in a straight line:
change of displacement
linear velocity =
change of time
i.e.
v = s t
(1)
The unit of linear velocity is metres per second (m/s).
−4
−2
2
0
−2
−3
−4
2
r = 4
4
6 x
Angular velocity
The speed of revolution of a wheel or a shaft is
usually measured in revolutions per minute or revolutions
per second but these units do not form part
of a coherent system of units. The basis in SI units
is the angle turned through in one second.
Angular velocity is defined as the rate of change of
angular displacement θ, with respect to time t. For an
object rotating about a fixed axis at a constant speed:
−8
angular velocity =
angle turned through
time taken
Figure 14.11
i.e.
ω = θ t
(2)