Drawing curves69P1^
2
An important feature of these curves is that they approach both the x and the y
axes ever more closely but never actually reach them. These lines are described as
asymptotes to the curves. Asymptotes may be vertical (e.g. the y axis), horizontal,
or lie at an angle, when they are called oblique.
Asymptotes are usually marked on graphs as dotted lines but in the cases above
the lines are already there, being co-ordinate axes. The curves have different
branches which never meet. A curve with different branches is said to be
discontinuous, whereas one with no breaks, like y = x^2 , is continuous.
The circle
You are of course familiar with the circle, and have probably done calculations
involving its area and circumference. In this section you are introduced to the
equation of a circle.
The circle is defined as the locusof all the points in a plane which are at a fixed
distance (the radius) from a given point (the centre). (Locus means path.)
As you have seen, the length of a line joining (x 1 , y 1 ) to (x 2 , y 2 ) is given by
length = ()xx 21 −+^2 ()yy 21 −^2.This is used to derive the equation of a circle.
In the case of a circle of radius 3, with its centre at the origin, any point (x, y) on
the circumference is distance 3 from the origin. Since the distance of (x, y) from
(0, 0) is given by ()xy−+ 002 ()−^2 , this means that ()xy−+ 002 ()−^2 = 3 or
x^2 + y^2 = 9 and this is the equation of the circle.
This circle is shown in figure 2.33.
These results can be generalised to give the equation of a circle centre (0, 0),
radius r as follows:
x^2 + y^2 = r^2xyO(^3) y
x
(x, y)
x^2 + y^2 = 3^2
(^4) (y – 5)
(9, 5) (x – 9)
Figure 2.33