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5.09 Graphs of circles

Worksheet
Graphs of circles
1

State the equations of the circles below:

a
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
y
b
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
2

For each of the following circles:

i

Write the coordinates of the centre.

ii

Find the radius of the circle.

iii

Find the x-intercepts.

iv

Find the y-intercepts.

v

Sketch the graph of the circle.

a
x^{2} + y^{2} = 25
b
x^{2} + y^{2} = 64
c
x^{2} + y^{2} = 16
d
x^{2} + y^{2} = 10
3

For each of the following circles:

i

Write the equation of the circle.

ii

Sketch the graph of the circle.

a

A circle that has its centre at the origin and a radius of 9 units.

b

A circle that has its centre at the origin and a radius of 4 units.

c

A circle that has its centre at the origin and a radius of \sqrt{18} units.

d

A circle that has its centre at the origin and a radius of \sqrt{3} units.

e

A circle that has its centre at the origin and a radius of 7 units.

4

Consider the circle with equation x^{2} + y^{2} = 121.

a

Find the diameter of the circle.

b

Find the y-values of the points on the circle that have an x-coordinate of -6.

5

Consider the circle with centre (0, 0) that passes through the point (5, - 10).

a

Find the radius of the circle.

b

Find the equation of the circle.

6

A circle with centre \left(0, 0\right) has an x-intercept at \left(6, 0\right).

a

Find the equation of the circle.

b

Find the exact area enclosed by the circle.

7

Does the point (- 4, 2) lie inside, outside or on the circle x^{2} + y^{2} = 21?

8

Does the point ( 3, 5) lie inside, outside or on the circle x^{2} + y^{2} = 5?

9

Does the point (0, -7) lie inside, outside or on the circle x^{2} + y^{2} = 49?

10

A circle has the equation 25 x^{2} + 25 y^{2} = 400.

a

Find the coordinates for the centre of the circle.

b

State the radius of the circle.

c

Hence, sketch the graph of the circle: 25 x^{2} + 25 y^{2} = 400.

11

A circle has the equation 3 x^{2} + 3 y^{2} - 48 = 0.

a

Find the coordinates for the centre of the circle.

b

State the radius of the circle.

c

Hence, sketch the graph of the circle: 3 x^{2} + 3 y^{2} - 48 = 0.

Graphs of translated circles
12

For each of the following pairs of graphs, describe how the graph in Diagram 1 is translated to the position shown in Diagram 2:

a

Diagram 1

-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
y

Diagram 2

-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
y
b

Diagram 1

-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
y

Diagram 2

-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
y
c

Diagram 1

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

Diagram 2

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
d

Diagram 1

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y

Diagram 2

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
13

How many units away from the origin have the circles below been translated?

a
-4
-3
-2
-1
1
2
3
4
x
1
2
3
4
5
6
7
8
9
10
y
b
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
c

\left(x + 5\right)^{2} + y^{2} = 4

d

x^{2} + \left(y - 7\right)^{2} = 6^{2}

14

Consider the graph of the circle below:

a

Describe the translation to get from x^{2} + y^{2} = 5^{2} to the circle shown.

b

State the equation of the circle shown in the graph.

-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
1
2
3
4
5
6
7
8
9
10
11
12
13
14
y
15

Find the coordinates of the new center after a circle centered at the origin is translated as follows:

a

2 units downwards.

b

7 units upwards.

c

3 units to the left.

d

6 units to the right.

16

Write down the equation of the new circle after x^{2} + y^{2} = 49 is translated:

a

5 units upwards.

b

5 units downwards.

c

5 units to the right.

d

5 units to the left and 6 units upwards.

17

A circle of radius 7 which was initially centred at the origin has been translated 6 units downwards.

a

State the radius of the resulting circle.

b

State the centre of the resulting circle.

c

Write the equation of the resulting circle.

18

Given the graph of x^{2} + y^{2} = 6^{2}, sketch the graph of x^{2} + \left(y - 3\right)^{2} = 6^{2}.

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
19

Given the graph of x^{2} + y^{2} = 4^{2}, sketch the graph of x^{2} + \left(y + 6\right)^{2} = 4^{2}.

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
20

A circle of radius 7, centred at the origin, is translated 9 units vertically downwards.

Write the equation of the resulting circle.

21

Graph the following translated circles as follows:

a

The circle x^{2} + y^{2} = 4^{2} is translated 4 units down.

b

The circle x^{2} + y^{2} = 25 is translated horizontally 4 units to the left.

22

Consider the circle with equation \left(x - 3\right)^{2} + y^{2} = 36.

a

Find the coordinates of the centre.

b

State the radius of the circle.

23

Graph the circles defined by the following equations:

a

x^{2} + \left(y - 2\right)^{2} = 4^{2}

b

x^{2} + \left(y + 6\right)^{2} = 5^{2}

c

\left(x - 3\right)^{2} + y^{2} = 25

d

\left(x + 5\right)^{2} + y^{2} = 16

e

\left(x - 4\right)^{2} + y^{2} = 9

f

\left(x + 4\right)^{2} + y^{2} = 25

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