Class XI

Solve Inequalities involving Circles

Lesson

The relation $\left(x-a\right)^2+\left(y-b\right)^2=r^2$(x−a)2+(y−b)2=r2, when graphed, shows a circle with centre $\left(a,b\right)$(a,b) and radius $r$r.

The inequality $\left(x-a\right)^2+\left(y-b\right)^2\le r^2$(x−a)2+(y−b)2≤r2 is also a relation (with exactly the same domain and range as the circle) but includes ordered pairs on and within the circle.

The inequality $\left(x-a\right)^2+\left(y-b\right)^2\ge r^2$(x−a)2+(y−b)2≥r2 similarly includes all ordered pairs on and outside the circle.

The ordered pairs on the circle are not included if the inequality is strictly less than, or strictly greater than $r^2$r2.

We offer two examples to illustrate the concept:

Example 1

Sketch the region given by the relation $x^2+\left(y-1\right)^2<16$x2+(y−1)2<16.

The boundary of the inequality, which is excluded from the relation, is a circle with centre $\left(0,1\right)$(0,1) and radius 4. The region includes all points within the circle as shown here:

Example 2

Identify the region defined by the following:

Note firstly that the circle is included in the region of interest but the line is not.

The intersection of the boundaries can be found by solving simultaneously the circle given by $x^2+y^2=25$x2+y2=25 and the line given by $y=3x-5$y=3x−5. We do this by substitution so that:

$x^2+\left(3x-5\right)^2$`x`2+(3`x`−5)2	$=$=	$25$25
$x^2+9x^2-30x+25$`x`2+9`x`2−30`x`+25	$=$=	$25$25
$10x^2-30x$10`x`2−30`x`	$=$=	$0$0
$10x\left(x-3\right)$10`x`(`x`−3)	$=$=	$0$0
$\therefore$∴ $x$`x`	$=$=	$0,3$0,3

At $x=0$x=0, $y=-5$y=−5 and at $x=3$x=3, $y=4$y=4, verifiable by substitution into either equation.

The circle and the line can be sketched, as shown below. Note that the shaded area is the area of exclusion, as is the custom of many school texts. Because there are two methods of identification possible (either shading in the region itself, or shading out anything other than the region) it is always a good idea to indicate with an arrow the region of interest as shown in the diagram.

Practice questions

Question 1

Determine whether the following points lie inside, outside, or on the circle $x^2+y^2=117$x2+y2=117.

$\left(-8,5\right)$(−8,5)
Inside
A
Can't say
B
On
C
Outside
D
$\left(-11,11\right)$(−11,11)
Outside
A
Inside
B
On
C
Can't say
D
$\left(0,0\right)$(0,0)
Inside
A
Outside
B
Can't say
C
On
D

Question 2

Consider the region $x^2+y^2\le25$x2+y2≤25.

Plot the boundary of the region satisfying the above inequality.

Loading Graph...
Select the region which satisfies the above inequality.
A
B
C
D

Question 3

Consider the region $x^2+y^2>4$x2+y2>4.

Plot the boundary of the region satisfying the above inequality.

Loading Graph...
Does $\left(1,1\right)$(1,1) satisfy the inequality?
Yes
A
No
B
Select the region which satisfies the above inequality.
A
B
C
D

Outcomes

11.CG.CS.1

Sections of a cone: Circles, ellipse, parabola, hyperbola, a point, a straight line and pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola. Standard equation of a circle.