Circles
The equation of a circle centred at (𝑎, 𝑏) with radius 𝑟 is given as
(𝑥 − 8𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2
The equation of a circle can be obtained given the following information:
(a) centre and radius
(b) centre and point on circumference – find the radius
(c) end points of the diameter – find the centre and radius
Expanded form of the equation of a circle: 𝑥 2 + 𝑦 2 + 𝑃𝑥 + 𝑄𝑦 + 𝑐 = 0
Tangents and Normals
Given a point on the circumference 𝑃:
The normal to the circle at the point P passes through the centre of the circle
The tangent to the circle at the point 𝑃 is perpendicular to the normal
𝑚𝑁 =
𝑦2 − 𝑦1
𝑥2 − 𝑥1
𝑚𝑇 = −
1
𝑚𝑁
Equation of Straight Line
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
NOTE: Use point on circumference to find equation of tangent or normal
Fidel DaSilva – CSEC Additional Mathematics
1) Find the equation of the circle centred at (3, −4) with radius of 2√3 units
(𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2
(𝑥 − 3)2 + (𝑦 + 4)2 = (2√3)
2
(𝑥 − 3)2 + (𝑦 + 4)2 = 12
2) Find the equation of the circle centred at (5,8) and passing through the point (9,10)
𝑟 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 = √(9 − 5)2 + (10 − 8)2
𝑟 = √(4)2 + (2)2 = √16 + 4 = √20
(𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2
(𝑥 − 5)2 + (𝑦 − 8)2 = 20
3) The endpoints of the diameter of a circle are P(-5,4) and Q(9,12). Find the equation of the
circle.
𝐶=(
𝑥1 + 𝑥2 𝑦1 + 𝑦2
−5 + 9 4 + 12
)=(
) = (2,8)
,
,
2
2
2
2
𝑟 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 = √(9 − 2)2 + (12 − 8)2
𝑟 = √49 + 16 = √65
(𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2
(𝑥 − 2)2 + (𝑦 − 8)2 = 65
Fidel DaSilva – CSEC Additional Mathematics
4) (a) Find the equation of the circle centred at C(2,5) and passing through the point P(6,8),
giving your answer in the form 𝑥 2 + 𝑦 2 + 𝑓𝑥 + 𝑔𝑦 + 𝑐 = 0
(b) Find the equation of the tangent to the circle at the point P
(a)
𝑟 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 = √(6 − 2)2 + (8 − 5)2 = √16 + 9 = √25 = 5
(𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2
(𝑥 − 2)2 + (𝑦 − 5)2 = 25
𝑥 2 − 4𝑥 + 4 + 𝑦 2 − 10𝑦 + 25 − 25 = 0
𝑥 2 + 𝑦 2 − 4𝑥 − 10𝑦 + 4 = 0
(b)
𝑦2 − 𝑦1 8 − 5 3
=
=
𝑥2 − 𝑥1 6 − 2 4
𝑚𝑁 =
𝑚𝑇 = −
1
4
=−
𝑚𝑁
3
𝑦 = 𝑚𝑥 + 𝑐
4
8 = − (6) + 𝑐 → 𝑐 = 16
3
4
𝑦 = − 𝑥 + 16
3
Fidel DaSilva – CSEC Additional Mathematics
5) (a) Find the equation of the circle centred at C(−4,5) and passing through the point
P(3,9), giving your answer in the form 𝑥 2 + 𝑦 2 + 𝑓𝑥 + 𝑔𝑦 + 𝑐 = 0
(b) Find the equation of the normal to the circle at the point P
(a)
𝑟 = √(−4 − 3)2 + (5 − 9)2 = √49 + 16 = √65
(𝑥 + 4)2 + (𝑦 − 5)2 = 65
𝑥 2 + 8𝑥 + 16 + 𝑦 2 − 10𝑦 + 25 − 65 = 0
𝑥 2 + 𝑦 2 + 8𝑥 − 10𝑦 − 24 = 0
(b)
𝑚𝑁 =
5−9
4
=
−4 − 3 7
4
51
9 = (3) + 𝑐 → 𝑐 =
7
7
4
51
𝑦= 𝑥+
7
7
Fidel DaSilva – CSEC Additional Mathematics
6) (a) The endpoints of the diameter of a circle are A(−2,3) and B(8, 13). Find the equation
of the circle in the form 𝑥 2 + 𝑦 2 + 𝑓𝑥 + 𝑔𝑦 + 𝑐 = 0
(b) Find the equation of the tangent to the circle at the point A
(a)
𝐶=(
𝑥1 + 𝑥2 𝑦1 + 𝑦2
−2 + 8 3 + 13
)=(
) = (3,8)
,
,
2
2
2
2
𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 = √(8 + 2)2 + (13 − 3)2 = √200
𝑟=
√200
2
(𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2
(𝑥 − 3)2 + (𝑦 − 8)2 = 50
𝑥 2 − 6𝑥 + 9 + 𝑦 2 − 16𝑦 + 64 − 50 = 0
𝑥 2 + 𝑦 2 − 6𝑥 − 16𝑦 + 23 = 0
(b)
𝑚𝑁 =
13 − 3
= 1 𝑚 𝑇 = −1
8+2
𝑦 = −𝑥 + 𝑐
3=2+𝑐 →𝑐 =1
𝑦 = −𝑥 + 1
Fidel DaSilva – CSEC Additional Mathematics
7) (a) The endpoints of the diameter of a circle are P(5, −7) and Q(9, 3). Find the equation of
the circle in the form 𝑥 2 + 𝑦 2 + 𝑓𝑥 + 𝑔𝑦 + 𝑐 = 0
(b) Find the equation of the normal to the circle at the point Q
(a)
𝐶=(
𝑥1 + 𝑥2 𝑦1 + 𝑦2
5 + 9 −7 + 3
)=(
) = (7, −2)
,
,
2
2
2
2
𝑟 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 = √(7 − 5)2 + (−2 − −7)2 = √4 + 25 = √29
(𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2
(𝑥 − 7)2 + (𝑦 + 2)2 = 29
𝑥 2 − 14𝑥 + 49 + 𝑦 2 + 4𝑦 + 4 − 29 = 0
𝑥 2 + 𝑦 2 − 14𝑥 + 4𝑦 + 24 = 0
(b)
𝑚𝑁 =
−2 − 3 −5 5
=
=
7−9
−2 2
𝑦 = 𝑚𝑥 + 𝑐
5
39
3 = (9) + 𝑐 → 𝑐 = −
2
2
5
39
𝑦= 𝑥−
2
2
Fidel DaSilva – CSEC Additional Mathematics
8) State the centre and radius of the circle (𝑥 − 3)2 + (𝑦 − 5)2 = 36
Centre: (3,5) Radius: 𝑟 = 6
9) State the centre and radius of the circle (𝑥 + 4)2 + (𝑦 − 9)2 = 20
Centre: (−4,9) Radius: 𝑟 = √20 = 2√5
10) State the centre and radius of the circle (𝑥 − 6)2 + (𝑦 + 10)2 = 75
Centre: (6, −10) Radius: 𝑟 = √75 = 5√3
11) State the centre and radius of the circle (𝑥 + 3)2 + (𝑦 + 7)2 = 8
Centre: (−3, −7) Radius: 𝑟 = √8 = 2√2
12) Find the centre and radius of the circle 𝑥 2 + 𝑦 2 − 4𝑥 − 6𝑦 + 8 = 0
[𝑥 2 − 4𝑥] + [𝑦 2 − 6𝑦] = 8
(𝑥 − 2)2 − 4 + (𝑦 − 3)2 − 9 = 8
(𝑥 − 2)2 + (𝑦 − 3)2 = 21
Centre: (2,3) Radius: 𝑟 = √21
13) Find the centre and radius of the circle 𝑥 2 + 𝑦 2 − 8𝑥 − 10𝑦 + 24 = 0
[𝑥 2 − 8𝑥] + [𝑦 2 − 10𝑦] = −24
(𝑥 − 4)2 − 16 + (𝑦 − 5)2 − 25 = −24
(𝑥 − 4)2 + (𝑦 − 5)2 = 17
Centre: (4,5) Radius: 𝑟 = √17
Fidel DaSilva – CSEC Additional Mathematics
14) Find the centre and radius of the circle 𝑥 2 + 𝑦 2 + 6𝑥 + 12𝑦 − 10 = 0
[𝑥 2 + 6𝑥] + [𝑦 2 + 12𝑦] = 10
(𝑥 + 3)2 − 9 + (𝑦 + 6)2 − 36 = 10
(𝑥 + 3)2 + (𝑦 + 6)2 = 55
Centre: (−3, −6) Radius: 𝑟 = √55
15) Find the centre and radius of the circle 𝑥 2 + 𝑦 2 − 14𝑥 + 10𝑦 − 5 = 0
[𝑥 2 − 14𝑥] + [𝑦 2 + 10𝑦] = 5
(𝑥 − 7)2 − 49 + (𝑦 + 5)2 − 25 = 5
(𝑥 − 7)2 + (𝑦 + 5)2 = 79
Centre: (7, −5) Radius: 𝑟 = √79
16) Find the centre and radius of the circle 𝑥 2 + 𝑦 2 + 12𝑥 − 6𝑦 + 20 = 0
[𝑥 2 + 12𝑥] + [𝑦 2 − 6𝑦] = −20
(𝑥 + 6)2 − 36 + (𝑦 − 3)2 − 9 = −20
(𝑥 + 6)2 + (𝑦 − 3)2 = 25
Centre: (−6,3) Radius: 𝑟 = 5
Fidel DaSilva – CSEC Additional Mathematics
17) (a) Find the centre and radius of the circle 𝑥 2 + 𝑦 2 − 8𝑥 − 16𝑦 + 15 = 0
(b) Find the equation of the tangent to the circle at the point (8,1)
(a)
[𝑥 2 − 8𝑥] + [𝑦 2 − 16𝑦] = −15
(𝑥 − 4)2 + (𝑦 − 8)2 − 16 − 64 = −15
(𝑥 − 4)2 + (𝑦 − 8)2 = 65
Centre: (4,8) Radius: 𝑟 = √65
(b)
𝑚𝑁 =
𝑦2 − 𝑦1 1 − 8
7
=
=−
𝑥2 − 𝑥1 8 − 4
4
𝑚𝑇 = −
1
4
=
𝑚𝑁 7
4
𝑦 = 𝑥+𝑐
7
4
25
1 = (8) + 𝑐 → 𝑐 = −
7
7
4
25
𝑦= 𝑥−
7
7
Fidel DaSilva – CSEC Additional Mathematics
18) Find the points of intersection of the circle 𝑥 2 + 𝑦 2 − 8𝑥 = 4 and the line 2𝑥 − 𝑦 = 8
Points of Intersection – Solve simultaneous equations
𝑥 2 + 𝑦 2 − 8𝑥 = 4
2𝑥 − 𝑦 = 8
𝑦 = 2𝑥 − 8
𝑥 2 + (2𝑥 − 8)2 − 8𝑥 = 4
𝑥 2 + 4𝑥 2 − 32𝑥 + 64 − 8𝑥 − 4 = 0
5𝑥 2 − 40𝑥 + 60 = 0
𝑥 2 − 8𝑥 + 12 = 0
𝑥 2 − 2𝑥 − 6𝑥 + 12 = 0
𝑥(𝑥 − 2) − 6(𝑥 − 2) = 0
(𝑥 − 2)(𝑥 − 6) = 0
𝑥 = 2, 𝑥 = 6
𝑥 = 2 → 𝑦 = 2(2) − 8 = −4
𝑥 = 6 → 𝑦 = 2(6) − 8 = 4
Points of Intersection: (2, −4) and (6,4)
Fidel DaSilva – CSEC Additional Mathematics
Practice
1) (a) Find the equation of the circle centred at (3, −4) and passing through the point T(5, 1)
in the form 𝑥 2 + 𝑦 2 + 𝑓𝑥 + 𝑔𝑦 + 𝑐 = 0
(b) Find the equation of the tangent to the circle at the point T
2) (a) Find the equation of the circle whose endpoints on one diameter are P(4,7) and
Q(6,13) in the form 𝑥 2 + 𝑦 2 + 𝑝𝑥 + 𝑞𝑦 + 𝑐 = 0
(b) Find the equation of the normal to the circle at the point Q
3) (a) State the centre and radius of the circle (𝑥 + 1)2 + (𝑦 − 4)2 = 50
(b) Find the equation of the tangent to the circle at the point (−6,9)
(c) Find the point of intersection between the circle and the line 𝑦 = 7 − 2𝑥
4) (a) Find the centre and radius of the circle 𝑥 2 + 𝑦 2 − 10𝑥 + 6𝑦 − 18 = 0
(b) Find the equation of the tangent to the circle at the point P(1,3)
5) (a) Find the equation of the circle whose endpoints on one diameter are A(5,6) and
B(3, −2)
(b) Find the equation of the tangent to the circle at the point B
6) (a) Find the centre and radius of the circle 𝑥 2 + 𝑦 2 − 16𝑥 − 12𝑦 + 20 = 0
(b) Find the equation of the normal to the circle at the point (4, −2)
(c) Find points of intersection of the circle and the line 2𝑥 − 𝑦 = 10
7) Find the equation of the tangent to the circle 𝑥 2 + 𝑦 2 + 4𝑥 − 6𝑦 + 3 = 0 at the point
(1,5)
8) The equation of a circle is given by 𝑥 2 + 𝑦 2 − 12𝑥 − 22𝑦 + 152 = 0
(a) Determine the coordinates of the centre of the circle and the length of the radius
(b) Determine the equation of the normal to the circle at the point (4, 10)
Fidel DaSilva – CSEC Additional Mathematics
9) The circle C has (–3, 4) and (1, 2) as endpoints of a diameter
(a) Show that the equation of C is 𝑥 2 + 𝑦 2 + 2𝑥 − 6𝑦 + 5 = 0
(b) Find the equation of the tangent to the circle at the point (−3,4)
10) The circle C is centred at (3, −1) and passes through the point 𝑃(7,2)
(a) Determine the equation of the circle in the form 𝑥 2 + 𝑦 2 + 𝑓𝑥 + 𝑔𝑦 + 𝑐 = 0
(b) Find the equation of the tangent to the circle at the point P(7, 2)
11) A circle C is defined by the equation 𝑥 2 + 𝑦 2 − 6𝑥 − 4𝑦 + 4 = 0
(a) Determine the centre and radius of the circle, C
(b) Find the equation of the normal to the circle C at the point (6, 2)
12) The equation of a circle, C, is given by 𝑥 2 + 𝑦 2 − 4𝑥 − 6𝑦 + 3 = 0
Find the equation of the tangent to C at the point (5, 2)
13) The equations of the circle, C, and the line, L, are given by 𝑥 2 + 𝑦 2 − 2𝑥 + 2𝑦 + 1 =
0 and 𝑦 = 𝑥 − 3 respectively.
(a) Determine the centre and radius of the circle, C
(b) Determine the points of intersection of the circle, C, and the straight line, L
14) The equation of a circle C is 𝑥 2 + 𝑦 2 + 4𝑥 − 2𝑦 − 20 = 0
(a) Determine the radius and the coordinates of the centre of C
(b) Determine the points of intersection of circle C and the line 𝑥 + 𝑦 = 4
Fidel DaSilva – CSEC Additional Mathematics
Answers
2
1) (a) 𝑥 2 + 𝑦 2 − 6𝑥 + 8𝑦 − 4 = 0 (b) 𝑦 = 5 𝑥
2) (a) 𝑥 2 + 𝑦 2 − 10𝑥 − 20𝑦 + 115 = 0 (b) 𝑦 = 3𝑥 − 5
3) (a) Centre: (−1,4) Radius: 𝑟 = √50 = 5√2 (b) 𝑦 = 𝑥 + 3 (c) (2, −11) and (4, −1)
2
7
4) (a) Centre: (5, −3) Radius: 𝑟 = 4 (b) 𝑦 = 3 𝑥 + 3
1
11
5) (a)(𝑥 − 4)2 + (𝑦 − 2)2 = 21 (b) 𝑦 = 4 𝑥 − 4
6) (a) Centre: (8,6) Radius: 𝑟 = √80 = 4√5 (b) 𝑦 = 2𝑥 − 10 (c) (4, −2) and (12,14)
7) 2𝑦 = 13 − 3𝑥
8) (a) Centre: (6,11) Radius: 𝑟 = √6
(b) 𝑥 − 2𝑦 + 16 = 0
9) (a) 𝑥 2 + 𝑦 2 + 2𝑥 − 6𝑦 + 5 = 0 (b) 𝑦 = 2𝑥 + 10
10) (a) 𝑥 2 + 𝑦 2 − 6𝑥 + 2𝑦 − 15 = 0 (b) 4𝑥 + 3𝑦 = 34
11) (a) Centre: (3, 2) Radius: 𝑟 = 3 (b) 𝑦 = 2
12) 3𝑥 − 𝑦 − 13 = 0
13) (a) Centre(1, −1) Radius: 𝑟 = 1 (b) (1, −2), (2, −1)
14) (a) Centre: (−2,1) Radius: 𝑟 = 5 (b) (2, −6), (3,1)
Fidel DaSilva – CSEC Additional Mathematics