The circle 237
x
2
+ y
2
+ 8x − 2y + 8 = 0 is of the form shown in
equation (2),
where a = −
8
2
= −4, b = −
−2
2
= 1
and
r =
(−4)
2
+ 1 2 − 8 =
√
9 = 3.
Hence, x 2 + y 2 + 8x − 2y + 8 = 0 represents a circle
centre (−4, 1) and radius 3, as shown in Figure 26.17.
a = Ϫ4
b = 1
Ϫ2
2
4
y
Ϫ4
Ϫ6
Ϫ8
0
r =
3
x
Figure 26.17
Alternatively, x 2 + y 2 + 8x − 2y + 8 = 0 may be rearranged as
(x + 4)
2
+ (y − 1)
2
− 9 = 0
i.e.
(x + 4)
2
+ (y − 1)
2
= 3
2
which represents a circle, centre (−4, 1) and radius 3,
as stated above.
Problem 19. 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 + 2 f y + c = 0
From above a = −
2e
2
, b = −
2 f
2
and r =
√
a 2 + b 2 − c
Hence, if x 2 + y 2 − 4x + 6y − 3 = 0
then
a = −
−4
2
= 2, b = −
6
2
= −3
and
r =
2 2 + (−3)
2
− (−3) =
√
16 = 4
Thus, the circle has centre (2, −3) and radius 4, as
shown in Figure 26.18.
Ϫ4
Ϫ2
2
3
4
1
y
Ϫ4
Ϫ5
Ϫ7
Ϫ8
Ϫ3
Ϫ2
2
4
6 x
0
r = 4
Figure 26.18
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.
Now try the following Practice Exercise
Practice Exercise 104 The equation of a
circle (answers on page 351)
1. Determine (a) the radius and (b) the
co-ordinates of the centre of the circle given
by the equation x 2 + y 2 − 6x + 8y + 21 = 0.
2. Sketch the circle given by the equation
x
2
+ y
2
− 6x + 4y − 3 = 0.
3. Sketch the curve x
2
+ (y − 1)
2
− 25 = 0.
4. Sketch the curve x = 6
1 −
y
6
2