1. Circle geometry: Equations / problems

2. Circle Geometry Equations
KUS objectives
BAT Find and use the general equations of circles
BAT solve geometry problems using your knowledge of circles
Starter: (next page)

3. Substitute y = 3x – 10 into equation of circle
WB 16
Show that the line y = 3x – 10 is a tangent to the circle 𝑥 2 + 𝑦 2 =10
Substitute y = 3x – into equation of circle
𝑥 2 + (3𝑥−10) 2 =10
10𝑥 2 −60𝑥+90=0
𝑥 2 −6𝑥+9=0
(𝑥−3) 2 =0
Since this has a repeated root there is only one point where the line meets
The circle. So the line is a tangent to the circle

4. The equation of a circle is 𝑥 2 + 𝑦 2 −4𝑥+2𝑦=15
WB 17
The equation of a circle is 𝑥 2 + 𝑦 2 −4𝑥+2𝑦=15
a) Find the coordinates of the centre C of the circle, and the radius of the circle
b) Show that the point P (4, -5) lies on the circle.
c) Find the equation of the tangent to the circle at the point P.
a) Rearrange to (𝑥−2) 2 + (𝑦+1) 2 −4−1=15
(𝑥−2) 2 + (𝑦+1) 2 =20
Center 2, − radius =2 5
b) Substituting values of point P gives (4) 2 + (−5) 2 − −5
=16+26 −16 −10= QED
c) Gradient of CP is −1−−5 2−4 =−2 so gradient of tangent is 1 2
Equation of tangent through P is 𝑦−−5= 1 2 (𝑥−4)
Or 𝑥−2𝑦=14

5. AB is the diameter of a circle. A is 1, 3 B is 7, −1
WB 18a
AB is the diameter of a circle. A is 1, 3 B is 7, −1
Find the coordinates of the centre C of the circle
Find the radius of the circle
Find the equation of the circle.
The line y + 5x = 8 cuts the circle at A and again at a second point D.
Calculate the coordinates of D [4]
e) Prove that the line AB is perpendicular to the line CD.
a) C is the midpoint AB to C (4, 1)
b) Using Pythagoras distance CA (4−1) 2 + (1−3) 2 = 13
c) Using centre and radius (𝑥−4) 2 + (𝑦−1) 2 = 13
d) Substituting 𝑦=8−5𝑥 into equation of circle gives
(𝑥−4) 2 + (8−5𝑥−1) 2 = 13
→ 𝑥 2 −8𝑥+16 +( 25 2 −70𝑥+49)= 13
→ 𝑥 2 −78𝑥+52= 0
→ 𝑥 2 −3𝑥+2= 0

6. AB is the diameter of a circle. A is 1, 3 B is 7, −1
WB 18b
AB is the diameter of a circle. A is 1, 3 B is 7, −1
Find the coordinates of the centre C of the circle
Find the radius of the circle
Find the equation of the circle.
The line y + 5x = 8 cuts the circle at A and again at a second point D.
Calculate the coordinates of D [4]
e) Prove that the line AB is perpendicular to the line CD.
d) Substituting 𝑦=8−5𝑥 into equation of circle gives
→ 𝑥 2 −3𝑥+2= 0
→ (𝑥−1)(𝑥−2)= 0
x-value at point D is x= 2 and so y=8−5 2 =−2
point D is (2, −2)
e) Gradient AB is 3−−1 1−7 =− 2 3
Gradient CD is 1−−2 4−2 =+ 3 2
product of gradients is − 2 3 × 3 2 =−1 so AB and CD are perpendicular

7. One thing to improve is –
KUS objectives BAT Find and use equations of circles
BAT find the centre and radius of a circle from its equation
self-assess
One thing learned is –
One thing to improve is –

8. END