The Circle
Higher Maths
The Circle
Information on the items below
General equation of a circle – completing the square
General equation of a circle
Introductory examples
Circle examples 2
Exam level questions
Tangents to circles 1
Tangents to circles 2
Circle examples 1
Circle examples 3
Basic Skills questions – exam level
Problem solving questions 1 – exam level
Problem solving questions 2 – exam level
Circle:- General equation ; completing the square.
In each example, by completing the square on the x and the y terms
and rearranging, show that the equations represent circles and state
the centre and the radius of each.
1
2
3
4
5
6
7
x2 + y2 – 12x – 4y – 31 = 0
x2 + y2 + 4x - 2y + 4 = 0
x2 + y2 – 6x – 8y - 3 = 0
x2 + y2 – 2x – 2y – 2 = 0
x2 + y2 – 10y - 24 = 0
x2 + y2 + 2x - 2y + 1 = 0
x2 + y2 + 20x – 20y + 119 = 0
8
9
10
11
12
x2 + y2 + 14x – 8y + 49 = 0
x2 + y2 – 16x – 6y + 59 = 0
x2 + y2 + 8x - 8y + 31 = 0
x2 + y2 – 4x - 32 = 0
x2 + y2 - 6x - 4y + 14 = 0
General Equation of a Circle
Check which of these equations represent circles.
Where they do, find the centre and the radius of the circle.
1
2
3
4
5
6
7
8
9
10
11
12
x2 + y2 – 6x – 4y – 3 = 0
x2 + y2 – 2x + 6y + 6 = 0
x2 + y2 – 10x – 8y + 32 = 0
x2 + y2 – 4x – 21 = 0
x2 + y2 + 2x – 4y + 1 = 0
x2 + y2 – 18x + 2y – 18 = 0
x2 + y2 + 12x – 10y + 12 = 0
x2 + y2 – 6x – 6y + 9 = 0
x2 + y2 – 121 = 0
x2 + y2 – 10y + 21 = 0
x2 + y2 – 14x + 6y + 39 = 0
x2 + y2 + 2x + 4y + 7 = 0
Tangents to Circles - set 1
In each example check that the given point
does lie on the circle and hence
find the equation of the tangent to the circle at that point.
Ex.
1
2
3
4
5
6
7
8
9
10
Equation of circle
x2 + y2 = 25
x2 + y2 = 5
(x – 2)2 + (y – 3)2 = 10
x2 + y2 – 2x – 4y – 21 = 0
(x + 2)2 + y2 = 26
x2 + y2 – 2y – 1 = 0
x2 + y2 + 6x – 2y + 5 = 0
x2 + (y + 2)2 = 4
x2 + y2 - 8x + 2y + 4 = 0
x2 + y2 - 2x - 2y - 23 = 0
Point
(-3,4)
(1,2)
(5,4)
(6,3)
(3,1)
(1,2)
(-2,3)
(-2,-2)
(6,2)
(5,-2)
Tangents to a circle at a point on the circle - set 2
In each example show that the point given lies on
the circle and find the equation of the tangent
to the circle at that point.
Write the equation of the tangent in the form ax + by + c = 0
Question
Equation of the circle
Point
1
x2 + y 2 = 5
(2,1)
2
x2 + y2 = 13
(3,2)
3
(x-2)2 + (y-1)2 = 13
(4,4)
4
(x+3)2 + (y-2)2 = 2
(-2,3)
5
(x-5)2 + (y-3)2 = 40
(7,9)
6
(x-2)2 + y2 = 45
(5,6)
7
x2 + y2 - 2x - 6y – 7 = 0
(2,7)
8
x2 + y2 + 2x + 8y – 41 = 0
(2,3)
9
x2 + y2 – 10x – 6y + 29 = 0
(6,5)
10
x2 + y2 – 4x – 4y + 3 = 0
(4,3)
11
x2 + y2 – 12y + 28 = 0
(2,8)
12
x2 + y2 – 8y + 3 = 0
(2,7)
13
x2 + y2 – 4x + 2y – 32 = 0
(3,5)
Circle Introductory examples
1.
State the centre and radius of these circles :
a) x2 + y2 = 49 b) 16x2 + 16y2 = 9 c) (x - 2)2 + (y - 3)2 = 1
d) x2 + (y + 6)2 = 36 e) (x + 2)2 + (y - 1)2 = 25
f) (x + 1)2 + (y + 1)2 = 100
2.
3.
Write down the equations of circles with the following centres and radii :
a) (2, 5) , r = 4 b) (-2, 7) , r = 5 c) (6, -2) , r = 3
a) Find the equation of the circle with centre the origin and passing through the
point (-1, -6).
b) Find the equation of the circle with centre (5, 4) and passing through (2, -1).
4. a) Find the equation of the circle passing through the points (0, 2), (2, 1) and (-1, 0).
b) Find the equation of the circle passing through the points (0, 0), (-1, 1) and (2, 0).
5.
If the point (h, 3) lies on the circle x2 + y2 + 4x - y - 2 = 0 , find h.
6. Show that the point A(-2, 3) lies on the circle x2 + y2 + x - 4y + 1 = 0
and find the equation of the tangent to the circle at A.
7.
a) Find the points of intersection of the line y = x - 2 and the circle
x2 + y2 + x + y - 6 = 0.
b) Find the points of intersection of the line y = 3x + 1 and the circle
x2 + y2 + 4x - 1 = 0.
8. Find the equation of the circle with centre C(2, -1) which passes through A(1, 1).
Find the equation of the tangent at A and show that this tangent
is also a tangent to the circle x2 + y2 -14x + 2y + 30 = 0.
9. Find the equation of the circle with centre P(5, -1) which passes through Q(-1, 2).
Find the equation of the tangent at Q and show that this tangent passes through
the centre of the circle x2 + y2 + 6x + 4y + 5 = 0.
10. Find the equation of the tangent to the circle x2 + y2 = 10 at the point A(1, -3).
Show that this line is also a tangent to the circle x2 + y2 - 4x - 8y - 20 = 0 and
find its point of contact.
11. i) Find the coordinates of the centre C and the radius
of the circle whose equation is
x2 + y2 - 2x - 4y - 3 = 0
ii) Show that the point A(3, 4) lies on the circle and find the equation
of the tangent at A.
iii) Show that the point P(7, 0) lies on this tangent.
iv) Find the equation of the circle which passes through the points C, A and P.
12. A and B are the points (3/5, -21/5) and (3, -3) respectively. Show
that A and B lie on the circle x2 + y2 - 4x + 8y + 18 = 0 and
find the equations of the tangents at A and B.
Solutions are on the next slide
Solutions
1.
a)
b)
c)
d)
e)
f)
centre
(0, 0)
(0, 0)
(2, 3)
(0, -6)
(-2, 1) (-1, -1)
radius
7
3/4
1
6
5
10
2
2
2
2
2
2
2. a) (x - 2) + (y - 5) = 16 b) (x + 2) + (y - 7) = 25 c) (x - 6) + (y + 2) = 9
3. a) x2 + y2 = 37 b) (x - 5)2 + (y - 4)2 = 34
4. a) x2 + y2 - x - y - 2 = 0
b) x2 + y2 - 2x - 4y = 0
5. h = - 2
6. 3x - 2y + 12 = 0
7. a) (2, 0) (-1, -3) b) (0, 1) (-1, -2)
8. (x -2)2 + (y + 1)2 = 5 ; x - 2y + 1 = 0
coincident point (5, 3)
9. (x - 5)2 + (y + 1)2 = 45
Tangent at Q has equation y = 2x + 4.
Centre (-3, -2) satisfies the equation.
10. Tangent at A has equation x = 3y + 10.
Coincident point (4, -2)
11. i) C(1, 2) r = 22
ii) tangent at A is y = - x + 7
iii) P(7, 0) satisfies the equation in ii)
iv) (x - 4)2 + (y - 1)2 = 10
12. Tangent at A has equation y = -7x
Tangent at B has equation y = -x
Circle Examples (1)
State the centre and the radius of each circle.
[ Centre (-g,-f) Radius =  ( g2 + f 2 – c ) ]
1
2
3
4
5
6
7
8
9
10
11
12
x2 + y2 –6x –4y + 4 = 0
x2 + y2 –2x –8y + 16 = 0
x2 + y2 + 4x + 4y + 4 = 0
x2 + y2 - 12x + 6y + 35 = 0
x2 + y2 -20x - 2y + 99 = 0
x2 + y2 - 8x = 0
x2 + y2 + 10y + 17 = 0
x2 + y2 + 2x - 2y + 1 = 0
x2 + y2 - 14x + 8y + 56 = 0
x2 + y2 + 16x + 61 = 0
x2 + y2 - 19 = 0
x2 + y2 - 24x - 22y + 263 = 0
Circle Examples (2)
1. In each example state the center and radius of the circle
(x – 3)2 + ( y – 4 )2 = 25
(x + 4)2 + ( y – 2 )2 = 9
(x + 8)2 + ( y + 5 )2 = 144
(x – 5)2 + y 2 = 16
x 2 + ( y + 1 )2 = 49
(x + 3)2 + ( y – 7 )2 = 100
(x – 2.5)2 + ( y – 4.5 )2 = 2.25
2. Write down the equations of the circles with the following centers and radius.
Centre
(2,5)
(3,-1)
(-6,-2)
(-4,5)
Radius
8
4
7
9
3. Do the following points lie in/on/outside the circle
with equation (x – 2)2 + ( y – 6 )2 = 34
a) (5,8) b) (-2,3) c) (7,9) d) (3,7)
4. Find the equation of the circle with center shown and passing
through the point stated.
Centre
Passing through the point
(2,5)
(3,7)
(-3,-4)
(2,6)
(4,1)
(4,4)
5. a) Find the equation of the tangent to the circle
(x – 3)2 + ( y – 4 )2 = 5
at the point (4,6) on the circle.
b) Find the equation of the tangent to the circle
(x + 1)2 + ( y –3)2 = 8
at the point (1,5) on the circle.
c) Find the equation of the tangent to the circle
(x – 2)2 + ( y – 1 )2 = 40
at the point (-4,-1) on the circle.
d) Find the equation of the tangent to the circle
x2 + ( y – 2 )2 = 17 at the point (4,3) on the circle.
e) Find the equation of the tangent to the circle
(x – 5)2 + y2 = 2
at the point (6,1) on the circle.
Circle Examples (3)
1. Write down the equations of the circles with center the origin and
radius a) 12 b) 3 c) 20 d) 2.5
2. Write down the radius of each of the circles
a) x2 + y2 = 81 b) x2 + y2 = 144
c) x2 + y2 = 12 d) x2 + y2 = 0.25
3. State whether the points below lie in/on/outside of the circle
with equation x2 + y2 = 60.
a) (7,4) b) (-3,8) c) (1,7)
4. Find the equations of the tangents to the circles at the point indicated.
y
y
 (4,3)
0
0
x
x
 (2,-5)
5. State the center and the radius of each of the circles
a) x2 + y2 –8x –10y + 4 = 0
b) x2 + y2 –x –y - 1 = 0
y
6. Find the equation of the circle shown in the diagram.
 (4,4)
0
x
7. Find the equation of the tangent to the circle
(x - 1)2 + ( y + 1 )2 = 8
at the point (3,1) on the circle.
8. Explain why the equation x2 + y2 –2x –4y + 26 = 0 does
not represent a circle ?
y
9. Find the equation of the circle shown in the diagram.
(5,0)

x
10. Show that the line y = 3x +10 is a tangent to the circle
x2 + y2 – 8x – 4y – 20 = 0 and find the point of contact.
11. Show that the line y = 2x – 5 is a tangent to the circle
x2 + y2 = 5 and find the point of contact.
12. Find the equation of the tangent to the circle x2 + y2 = 10
at the point (-3,-1) on the circle. Show that this tangent is also a tangent
to the circle x2 + y2 – 20y + 60 = 0 and find the
point of contact with this circle.
13. Show that the line x+9 = 0 is a tangent to the circles
x2 + y2 = 81 and x2 + y2 + 10x + 24y + 153 = 0 and find the points of contact.
Circle – Exam level questions and Past paper questions.
1. Find the possible values of ‘k’ for which the
line y = x – k is a tangent to the circle x2 + y2 = 18.
2. An ear ring is to be made from
silver wire and is designed in the shape
of two circles with tangents to the
outer circle as shown in the diagram
on the left.
The other diagram shows the ear
ring related to the coordinate axis.
The circles touch at (0,0).
The equation of the inner circle
is x2 + y2 + 3y = 0. The other circle
intersects the y axis at (0,-4).
The tangents meet the y axis at (0,-6).
Find the total length of wire needed to make
this ear ring.
-4
-6

14

3. The diagram shows a ‘gingerbread
man’ 14 cm high with a circular
head and body. The equation of
the body is
x2 + y2 -10x – 12y + 45 = 0
and the line of centres is parallel
to the y axis.
Find the equation of the head.
4. Show that the point (3,1) lies on the circle with equation
x2 + y2 – 4x + 6y – 4 = 0. Find the equation of the tangent to the
circle at this point.
5. AB is a tangent at B to the circle shown.
The circle centre C has equation
(x-2)2 + (y-2)2 = 25.
The point A has coordinates (10,8).
Find the area of triangle ABC
y
A
B
C
O
x
6. Show that the equation x2 + y2 + 2x + 3y + 5 = 0
does not represent a circle.
7. The straight line y = x cuts the circle
x2 + y2 -6x -2y -24 = 0 at the
points A and B.
Find the coordinates of A and B
Find the equation of the circle which
has AB as diameter.
y = x
y

A
O
B
8. Find the equation of the circle which has
P(-2,-1) and Q(4,5) as the end points of a diameter.
x
9. The diagram shows three circles.
The centres A, B and C are collinear i.e. they
All lie on the same straight line. The equation of
outer circles are
(x+12)2+(y+15)2=25 and (x-24)2+(y-12)2=100.
Find the equation of the central circle.
y
C
O
B
A
10. Find the equation of the tangent to the
circle x2 + y2 + 2x – 4y – 15 = 0 , at the point (3,4) on the circle.
11. The line y = -1 is a tangent to the circle which passes through
the points (0,0) and (6,0). Find the equation of this circle.
x
12. Show that the line x + 3y – 16 = 0 as a tangent to the circle
x2 + y2 + 10x + 6y – 56 = 0. Find the point of contact.
13. Find the coordinates of the points where the line y = 2x – 1 cuts
the circle with equation x2 + y2 = 2.
B
14. In the diagram, AB and AC are the tangents
from a point A(9,0) to the circle with equation
x2 + y2 = 16 with centre O. Find the area of
the kite AOBC.
A
O
C
15. Show that the line x + y = 10 is a tangent
to the circle x2 + y2 – 2x – 10y + 18 = 0 and find the
coordinates of the point of contact.
16. Find the centre and radius of the circle with equation
x2 + y2 -6x -8y = 0. Find the equation of the tangent to this
circle at the point (6,8) on the circle.
17. Show that the equation of the circle which passes through
the points (0,0), (4,0) and (0,-2) is x2 + y2 – 4x + 2y = 0.
Show that the line y = 2x – 10 is a tangent to this circle and
find the point of contact.
18. Find the equation of the tangent to the
circle x2 + y2 – 3x + y – 16 = 0 at the point (4,3).
y
19. Two identical circles touch at the point
P(9,3) as shown in the diagram. One of the
circles has equation x2 + y2 - 10x – 4y + 12 = 0.
Find the equation of the other circle.
 P(9,3)
O
20. Find the coordinates of the points of intersection
of the circle with equation x2 + y2 + 10x – 2y - 14 = 0 and the line with
equation y = 2x + 1.
x
21. The circle shown has equation
(x-3)2 + (y+2)2 = 25
Find the equation of the tangent to this circle
at the point (6,2). Where does this
tangent cut the y axis?
y
(6,2)
O

x
22. The line with equation x – 3y = k, is a tangent to the circle
with equation x2 + y2 – 6x + 8y + 15 = 0. Find the possible values of ‘k’.
23. Show that the points (5,4) , (-3,-2) and (5,-2) form a right angled triangle
and hence find the equation of the circle through these three points.
24. Show that the tangents to the circle x2 + y2 – 4x - 2y - 20 = 0 at
the points A(7,1) and B(-1,5) intersect at the point C(7,11).
25. Find the equation of the circle which has centre (2,3) and
which passes through the point (5,6).
a) Prove that the line x - y + 7 = 0 is a tangent to this circle.
b) Show that where this line meets the circle x2 + y2 = r2, x must
satisfy the equation 2x2 + 14x + (49-r2) = 0.
c) Find the value of ‘r’ if the line x - y + 7 = 0 is to be a tangent
to the circle x2 + y2 = r2.
Solutions are on the next slide
Solutions to circle problem solving – sheet 1
Solutions are on the next slide
Solutions to circle problem solving – sheet 2
Information on each item is shown below.
General equation of a circle – completing the square
This contains examples which start with a general circle form
and ends up with the (x-a)2+(y-b)2= r2
General equation of a circle
Starting from the general equation you have to work out the centre and radius
Tangents to circles (set 1)
Working out the equation of the tangent to a circle at a point on the circle
Tangents to circles (set 2)
More examples on working out the equation of the tangent
to a circle at a point on the circle.
Introductory examples
A set of examples using all the forms of the equations of the circle.
Continued on next slide
Circle Examples 1
Stating the centre and radius using the general equation.
Circle Examples 2
Simple examples using the basic ideas of the circle.
Circle Examples 3
More difficult examples using the basic ideas of the circle.
Exam level questions
A set of 25 exam level questions on the circle.
Basic skills questions
A set of basic skills questions on the circle. These questions are exam level.
Problem solving questions – 2 sets
Two sets of questions on the circle testing problem solving skills.
These questions are exam level.