2 Annuli between circles

Equations in $x$ and $y$ , such as (1) i.e. $x^2+y^2=R^2$ and (2) i.e. ${\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2=R^2$ for circles, define curves in the $Oxy$ plane. However, inequalities are necessary to define regions . For example, the inequality

$x^2+y^2<1$

is satisfied by all points inside the unit circle - for example $\left(0,0\right),\left(0,\frac{1}{2}\right),\left(\frac{1}{4},0\right),\left(\frac{1}{2},\frac{1}{2}\right)$ .

Similarly $x^2+y^2>1$   is satisfied by all points outside that circle such as $\left(1,1\right)$ .

Figure 31

Example 16

Sketch the regions in the $Oxy$ plane defined by

1. ${\left(x-1\right)}^2+y^2<1$
2. ${\left(x-1\right)}^2+y^2>1$

Solution

The equality ${\left(x-1\right)}^2+y^2=1$ is satisfied by any point on the circumference of the circle centre (1,0) radius 1. Then, remembering that ${\left(x-1\right)}^2+y^2$ is the square of the distance between any point $\left(x,y\right)$ and (1,0), it follows that

1. ${\left(x-1\right)}^2+y^2<1$ is satisfied by any point inside this circle (region (A) in the diagram.)
2. ${\left(x-1\right)}^2+y^2>1$ defines the region exterior to the circle since this inequality is satisfied by every point outside. (Region (B) on the diagram.)

   

The region between two circles with the same centre (i.e. concentric circles) is called an annulus or annular region . An annulus is defined by two inequalities. For example the inequality

$x^2+y^2>1$ (7)

defines, as we saw, the region outside the unit circle.

The inequality

$x^2+y^2<4$ (8)

defines the region inside the circle centre origin radius 2.

Hence points $\left(x,y\right)$ which satisfy both the inequalities (7) and (8) lie in the annulus between the two circles. The inequalities (7) and (8) are combined by writing

$1<x^2+y^2<4$

Figure 32

Task!

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