2 Annuli between circles

Equations in x and y , such as (1) i.e. x 2 + y 2 = R 2 and (2) i.e. ( x − x 0 ) 2 + ( y − y 0 ) 2 = R 2 for circles, define curves in the O x y 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 ( 0 , 0 ) , ( 0 , 1 2 ) , ( 1 4 , 0 ) , ( 1 2 , 1 2 ) .

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

Figure 31

A graph showing a shaded circular region centered at the origin, labeled with inequalities.

Long description
Example 16

Sketch the regions in the O x y plane defined by

  1. ( x − 1 ) 2 + y 2 < 1
  2. ( x − 1 ) 2 + y 2 > 1
Solution

The equality ( x − 1 ) 2 + y 2 = 1 is satisfied by any point on the circumference of the circle centre (1,0) radius 1. Then, remembering that ( x − 1 ) 2 + y 2 is the square of the distance between any point ( x , y ) and (1,0), it follows that

  1. ( x − 1 ) 2 + y 2 < 1 is satisfied by any point inside this circle (region (A) in the diagram.)
  2. ( x − 1 ) 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.)

    A graph on a Cartesian coordinate system showing a circle centered at (1, 0) with a radius of 1.

    Long description

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 ( x , y ) 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

The image shows a graph on a coordinate plane depicting an annulus, which is the region between two concentric circles.

Long description
Task!

Sketch the annulus defined by the inequalities

1 < ( x − 1 ) 2 + y 2 < 9

The quantity ( x − 1 ) 2 + y 2 is the square of the distance of a point ( x , y ) from the point (1,0). Hence, as we saw earlier, the left-hand inequality

1 < ( x − 1 ) 2 + y 2 which is the same as ( x − 1 ) 2 + y 2 > 1

is the region exterior to the circle C 1 centre ( 1 , 0 ) radius 1.

Similarly the right-hand inequality

( x − 1 ) 2 + y 2 < 9

defines the interior of the circle C 2 centre ( 1 , 0 ) radius 3. Hence the double inequality holds for any point in the annulus between C 1 and C 2 .

The image displays a graph with two concentric circles, \(C_1\) and \(C_2\), centered at the origin on the \(xy\)-plane.

Long description
Exercises
  1. Write down the radius and the coordinates of the centre of the circle for each of the following equations
    1. x 2 + y 2 = 16
    2. ( x − 4 ) 2 + ( y − 3 ) 2 = 12
    3. ( x + 3 ) 2 + ( y − 1 ) 2 = 25
    4. x 2 + ( y + 1 ) 2 − 4 = 0
    5. ( x + 6 ) 2 + y 2 − 36 = 0
  2. Obtain in each case the equation of the given circle
    1. centre C ( 0 , 0 ) radius 7
    2. centre C ( 0 , 2 ) radius 2
    3. centre C ( 4 , − 4 ) radius 4
    4. centre C ( − 2 , − 2 ) radius 4
    5. centre C ( − 6 , 0 ) radius 5
  3. Obtain the radius and the coordinates of the centre for each of the following circles
    1. x 2 + y 2 − 10 x + 12 y = 0
    2. x 2 + y 2 + 2 x − 4 y = 11
    3. x 2 + y 2 − 6 x − 16 = 0
  4. Describe the regions defined by each of these inequalities
    1. x 2 + y 2 > 4
    2. x 2 + y 2 < 16
    3. the inequalities in (i) and (ii) together
  5. State an inequality that describes the points that lie outside the circle of radius 4 with centre ( − 4 , 2 ) .
  6. State an inequality that describes the points that lie inside the circle of radius 6 with centre ( − 2 , − 1 ) .
  7. Obtain the equation of the circle which has centre ( 3 , 4 ) and which passes through the point ( 0 , 5 ) .
  8. Show that if A ( x 1 , y 1 ) and B ( x 2 , y 2 ) are at opposite ends of a diameter of a circle then the equation of the circle is ( x − x 1 ) ( x − x 2 ) + ( y − y 1 ) ( y − y 2 ) = 0.

    (Hint: if P is any point on the circle obtain the slopes of the lines A P and B P and recall that the angle in a semicircle must be a right-angle.)

  9. State the equation of the unique circle which touches the x − axis at the point (2,0) and which passes through the point ( − 1 , 9 ) .
    1. radius 4 centre ( 0 , 0 )
    2. radius 12 centre ( 4 , 3 )
    3. radius 5 centre ( − 3 , 1 )
    4. radius 2 centre ( 0 , − 1 )
    5. radius 6 centre ( − 6 , 0 )
    1. x 2 + y 2 = 49
    2. x 2 + ( y − 2 ) 2 = 4
    3. ( x − 4 ) 2 + ( y + 4 ) 2 = 16
    4. ( x + 2 ) 2 + ( y + 2 ) 2 = 16
    5. ( x + 6 ) 2 + y 2 = 25
    1. centre ( 5 , − 6 ) radius 61
    2. centre ( − 1 , 2 ) radius 4
    3. centre (3,0) radius 5
    1. the region outside the circumference of the circle centre the origin radius 2.
    2. the region inside the circle centre the origin radius 4 (often referred to as a circular disc)
    3. the annular ring between these two circles.
  5. ( x + 4 ) 2 + ( y − 2 ) 2 > 16
  6. ( x + 2 ) 2 + ( y + 1 ) 2 < 6
  7. ( x − 3 ) 2 + ( y − 4 ) 2 = 10
  8. ( x − x 1 ) ( x − x 2 ) + ( y − y 1 ) ( y − y 2 ) = 0.
  9. ( x − 2 ) 2 + ( y − 5 ) 2 = 25 (Note: since we are told the circle touches the x − axis at (2,0) the centre of the circle must be at the point ( 2 , y 0 ) where y 0 = R ).