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3.07 Circles and semi-circles

Worksheet
The equation of a circle with centre at the origin
1

Given the equation of a circle x^{2} + y^{2} = 64:

a
State the coordinates of the centre.
b
Find the radius of the circle.
c
Find the x-intercepts.
d
Find the y-intercepts.
e
Graph the circle.
2

For each of the following circles:

i
Plot the graph of the circle.
ii
Write the equation of the circle.
a
A circle with centre at the origin and a radius of 4 units.
b
A circle with centre at the origin and a radius of \sqrt{3} units.
3

Write down the equation for a circle with centre \left(0, 0\right) and radius 2.

4

The equation of a circle is given by x^{2} + y^{2} = 32. Calculate the radius of this circle, writing your answer in simplest surd form.

5

Consider the graph of the circle shown below.

State the following in interval notation.

a

The domain

b

The range

-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
x
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
y
6

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

a

Find the exact radius of the circle.

b

Find the equation of the circle.

7

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 inside the circle.

8

Consider the circle with equation x^{2} + y^{2} = 49.

a

What is the diameter of the circle?

b

Find the y-values of the points on the circle that have an x-coordinates of 1.

9

Sketch the graph of x^{2} + y^{2} = 3.

10

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

11

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

a

What are the coordinates of the centre of the circle?

b

State the radius of the circle.

c

Sketch the curve of 25 x^{2} + 25 y^{2} = 400.

12

Determine whether the following are equations of circles.

a

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

b

4 x^{2} + 5 y^{2} = 4

c

y^{2} = 2 + x^{2}

d

4 x^{2} + 4 y^{2} = 16

e

x^{2} + y^{2} = 4

f

y^{2} = 2 - x^{2}

The general equation of a circle
13

Consider a circle with centre at (3, 3) and radius of 6 units.

a
Plot the graph of the circle.
b
Write the equation of the circle.
14

Given the equation of a circle x^{2} + \left(y - 3\right)^{2} = 9:

a
State the coordinates of the centre
b
Find the radius of the circle
c
Find the x-intercept
d
Find the y-intercepts
e
Graph the circle
15

For each of the following circles:

i
Plot the graph
ii
State the domain
iii
State the range
a
x^{2} + \left(y - 4\right)^{2} = 4
b
\left(x + 5\right)^{2} + \left(y + 3\right)^{2} = 16
16

Find the equation of a circle that has been translated 2 units downwards from the origin with radius of 5 units.

17

Consider the circle with centre (8, 6) and radius 8.

a

Write down the equation of the circle.

b

Does the circle pass through (0, 0)?

18

Does \left(x - 7\right)^{2} + \left(y - 5\right)^{2} = - 4 represent the equation of a circle?

19

Is the following statement true or false?

The graph of \left(x - 4\right) + \left(y + 3\right) = 4 is a circle with radius 2 and centre at \left(4, - 3 \right).

20

The equation of a circle is given by \left(x - 6\right)^{2} + \left(y - 6\right)^{2} = 12.

a

Find the coordinates of the centre.

b

Find the radius, in simplest surd form.

21

Consider the circle on the graph.

a

Find the centre of the circle.

b

Find the radius of the circle.

c

What is the equation of the circle?

-6
-5
-4
-3
-2
-1
1
x
-6
-5
-4
-3
-2
-1
1
y
22

Write the equation of the circle given the following centre and radius.

a

Centre \left(1, - 3 \right) and radius of 5 units.

b

Centre \left( - 5.7 , 0\right) and radius of 3 units.

23

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

24

A circle centred at (7, - 8) has an x-intercept at (- 10, 0). Find the exact radius of the circle.

25

A circle is described by the following equation:

\left(x + \frac{1}{2}\right)^{2} + \left(y + \frac{1}{2}\right)^{2} = \frac{9}{4}

a

Find the centre of the circle.

b

Find the radius of the circle.

c

Plot the graph of the circle.

26

How many x-intercepts does the circle \left(x - 7\right)^{2} + \left(y - 1\right)^{2} = 4 have?

27

A circle has a domain of \left[2, 10\right] and a range of \left[ - 4 , 4\right].

a

Plot the circle.

b

State the equation for the circle

28

Consider the circle \left(x - 3\right)^{2} + \left(y + 1\right)^{2} = 9shown:

If this circle was inscribed (fitted exactly) inside a square so that the sides of the circle touched the sides of the square, state the coordinates of the following vertices of the square:

a

top-left vertex

b

top-right vertex

c

bottom-left vertex

d

bottom-right vertex

-1
1
2
3
4
5
6
7
x
-4
-3
-2
-1
1
2
3
4
y
Complete the square to convert the equation to general form
29

For each equation below, determine:

i

The equation of the circle in the form \left(x - h\right)^{2} + \left(y - k\right)^{2} = r^2.

ii

The coordinates of the centre of the circle.

iii

The radius of the circle.

a
x^{2} - 8 x + y^{2} - 20 = 0
b
x^{2} + 4 x + y^{2} + 6 y - 3 = 0
c
x^{2} - 2 x + y^{2} + 8 y - 19 = 0
d
x^{2} + 8 x + y^{2} - 6 y + 24 = 0
30

Consider the equation of a circle given by x^{2} + 4 x + y^{2} - 10 y = 52.

a

Rewrite the equation of the circle in the form \left(x - h\right)^{2} + \left(y - k\right)^{2} = r^2.

b

What are the coordinates of the centre of the circle?

c

What is the radius of the circle?

d

Graph the circle.

31

Consider the equation of a circle given by x^{2} + y^{2} - 2 x - 10 y - 24 = 0.

a

Rewrite the equation of the circle in the form \left(x - h\right)^{2} + \left(y - k\right)^{2} = r^2.

b

What are the coordinates of the centre?

c

Find the radius in simplest surd form.

d

Find the y-intercepts.

e

Find the x-intercepts.

f

Graph the circle.

32

Consider the equation of a circle given by y^{2} + 2 y + 8 = 12 x - x^{2} + 7.

a

Rewrite the equation of the circle in the form \left(x - h\right)^{2} + \left(y - k\right)^{2} = r^2.

b

What are the coordinates of the centre of the circle?

c

What is the radius of the circle?

d

Graph the circle.

33

A circle has the equation x^{2} + y^{2} - 32 x + 30 y + 462 = 0.

a

Rearrange the equation into the form \left(x - h\right)^{2} + \left(y - k\right)^{2} = r^{2}.

b

State the domain of the circle in interval notation.

c

State the range of the circle in interval notation.

d

Determine whether the following points lie inside or outside the circle.

i

\left(10, - 15 \right)

ii

\left(16, - 15 \right)

iii

\left(16, -12\right)

iv

\left(25, - 14 \right)

34

Graph the circle x^{2} + y^{2} - 6 x + 4 y = 3.

35

If a circle of radius 8 rolls along the x-axis, state an equation for the path of the centre of the circle.

-14
-12
-10
-8
-6
-4
-2
2
x
2
4
6
8
10
12
14
16
y
The equation of a semicircle with centre at the origin
36

For the following graphs of semicircles, find:

i
The centre
ii
The radius
iii
The equation
a
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
b
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
c
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
d
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
37

A semicircle has equation y = \sqrt{64 - x^{2}}.

a

State the centre of the semicircle.

b

State the radius of the semicircle.

c

Sketch the graph of y = \sqrt{64 - x^{2}}.

38

Is the function f \left( x \right) = - \sqrt{49 - x^{2}} one-to-one?

39

Consider the circle with equation x^{2} + y^{2} = 4.

a

Rearrange the equation to make y the subject.

b

Write down the equation of the semicircle which has a radius of 2 and negative y-values.

c

Draw the graph of the semicircle.

The general equation of a semicircle
40

Consider the semicircle below:

a

Determine the domain of the function.

b

Determine the range of the function.

-6
-4
-2
2
4
6
x
-6
-4
-2
2
4
6
y
41

For each equation of a semicircle below:

i
Sketch the graph described by the equation.
ii
State the domain.
iii
State the range.
a
y = \sqrt{4 - \left(x + 6\right)^{2}}
b
y = - \sqrt{9 - \left(x + 3\right)^{2}} - 4
c
y=\sqrt{36-\left(x-1\right)^2}
d
y=-\sqrt{1-(x+5)^2}
e
y=-\sqrt{16-(x-3)^2}
f
y=\sqrt{25-(x+6)^2}
42

Consider the graph of y = \sqrt{4 - x^{2}} shown.

Find the new equation if the semi-circle is translated:

a

Down by 7 units.

b

To the left by 3 units.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
43

The top half of a circle has a domain of \left[ - 10 , 2\right] and a range of \left[ - 2 , 4\right].

a

Plot the semicircle.

b

State the equation of the semicircle.

Application
44

A soccer match is being televised.

One of the cameras is mounted on a drone which is programmed to zoom in on the ball only when it is inside the centre circle.

The drone uses a coordinate system to track the position of the ball, where the origin is at the bottom left corner of the field and each unit corresponds to 1\text{ m}.

a

The centre of the centre circle is in the exact middle of the field.

What are the coordinates of the circle's centre?

b

What is the radius of the centre circle?

c

State the equation of the circle.

d

State the domain of the circle in interval notation.

e

State the range of the circle in interval notation.

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MA11-1

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