Topics
4 Nonlinear relationships
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AustraliaVIC
Level 10

4.04 Circles

Lesson
Worksheet
Practice
Worksheet
Graphs of circles
1

State the centre and radius of each of the following circles:

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

For each of the following equations of circles:

i

State the centre of the circle.

ii

State the radius of the circle.

a
\left(x - 5\right)^{2} + \left(y + 6\right)^{2} = 17
b
\left(x - 5\right)^{2} + \left(y - 5\right)^{2} = 162
c
\left(x - 0.4\right)^{2} + \left(y + 3.8\right)^{2} = 2
d
\left(x + 3.2\right)^{2} + \left(y - \dfrac{3}{2}\right)^{2} = \dfrac{4}{9}
3

For each of the following equations of circles:

i

State the the centre of the circle.

ii

State the radius of the circle.

iii

Sketch the graph of the circle.

a
\left(x - 2\right)^{2} + \left(y + 2\right)^{2} = 36
b

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

c
\left(x + 1\right)^{2} + \left(y - 3\right)^{2} = 16
d
\left(x + 3\right)^{2} + \left(y + 4\right)^{2} = 25
e
\left(x - \dfrac{3}{4}\right)^{2} + \left(y - \dfrac{3}{4}\right)^{2} = \dfrac{9}{4}
4

How many x-intercepts does the circle \left(x + 8\right)^{2} + \left(y + 1\right)^{2} = 1 have?

5

The circle \left(x + 1\right)^{2} + \left(y + 5\right)^{2} = 16 is inscribed (fitted exactly) inside a square with base parallel to the x-axes.

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.

Equations of circles
6

Find the equation of each of the following circles:

a
-3
-2
-1
1
2
3
4
5
6
7
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
b
-5
-4
-3
-2
-1
1
2
3
4
5
x
-1
1
2
3
4
5
6
7
8
9
y
c
-3
-2
-1
1
2
3
4
5
6
7
x
-7
-6
-5
-4
-3
-2
-1
1
2
3
y
d
-1
1
2
3
4
5
6
7
8
9
10
11
x
-1
1
2
3
4
5
6
7
8
9
10
11
y
7

Find the equation of a circle in standard form given the following centre and radius:

a

Centre: \left(7, -1\right); r=3

b

Centre: \left(-2, -3\right); r=5

c

Centre: \left(-4.3, 0\right); r=6

d

Centre: \left(-10, 2\right); r=12

8

Write down the equation of the new circle after x^{2} + y^{2} = 25 is translated:

a

10 units upwards.

b

10 units downwards.

c

10 units to the right.

d

10 units to the left and 4 units upwards.

9

A circle has its centre at \left(1, 2\right) and a radius of 4 units.

a

Sketch the graph of the circle.

b

Find the equation of the circle in standard form.

10

A circle has its centre at \left(3, - 2 \right) and a radius of 4 units.

a

Sketch the graph of the circle.

b

Find the equation of the circle in standard form.

11

Consider the circle with centre \left(- 8,- 6\right) and radius 10.

a

Find equation of the circle in standard form.

b

Does the circle pass through (0, 0)?

12

A circle centred at \left(2, 4\right) passes through the point \left(3, 3\right).

a

Find the exact radius of the circle.

b

Find the equation of the circle in standard form.

13

Find the equation of the circle that has its centre at \left( - 2 , 4\right) and passes through the point \left(3, 5\right).

14

A circle centred at (- 2,-5) has an x-intercept at (- 1,0). Find the exact radius of the circle.

15

For each of the following circles:

i

State the coordinates of the centre of the circle.

ii

Find the exact radius of the circle.

iii

Find the equation of the circle.

a

A circle with endpoints of the diameter at \left(10, 6\right) and \left( - 12 , - 14 \right). .

b

A circle with the smallest radius that contains the points \left(1, 2\right) and \left( - 5 , 4\right) within or on its boundary.

Circles in general form
16

For the circles described below, find the equation of the circle in general form:

a

A circle with centre \left(5,-2\right) and a radius of 3 units.

b

A circle with centre \left(-1,4\right) and a radius of 6 units.

17

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 standard form.

b

State the coordinates of the centre.

c

Find the radius.

d

Find the y coordinates of the y-intercepts.

e

Find the x coordinates of the x-intercepts.

f

Sketch the graph of the circle.

18

For each of the following equations of circles:

i

Rewrite the equation of the circle in standard form.

ii

State the coordinates of the centre of the circle.

iii

Find the radius of the circle.

iv

Sketch the graph the circle.

a
x^{2} + 8 x + y^{2} - 10 y = 8
b
x^{2} + 4 x + y^{2} + 6 y - 3 = 0
c
x^{2} - 4 x + y^{2} - 12 = 0
d
x^{2} - 18 x + y^{2} - 16 y + 64 = 0
e
y^{2} + 12 y + 4 = 16 x - x^{2} - 71
f
x^{2} + y^{2} - 4 x - 4 y = 17
g
4 x^{2} + 4 y^{2} + 12 x - 36 y + 9 = 0
19

Sketch the graph of the following equations:

a
x^{2} + y^{2} - 4 x + 2 y = 4
b
x^{2} + y^{2} + 5 x - y = 0
c
x^{2} + y^{2} - 3 x + 8 y = - \dfrac{37}{4}
d
x^{2} + y^{2} - x - \dfrac{5}{2}y = \dfrac{371}{16}
20

The following equation describes a circle:x^{2} + y^{2} - 11 x + 3 y + \dfrac{55}{2} = 0

a

Express the equation of the circle in standard form.

b

Find the radius of the circle.

c

The circle on the given graph has the same centre as the circle with the above equation, but not the same radius. Would the circle with equation \\x^{2} + y^{2} - 11 x + 3 y + \dfrac{55}{2} = 0 lie inside or outside the given circle?

-2
2
4
6
8
10
12
14
x
-8
-6
-4
-2
2
4
6
8
y
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VCMNA339

Explore the connection between algebraic and graphical representations of relations such as simple quadratic, reciprocal, circle and exponential, using digital technology as appropriate

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