1. Co-ordinate Geometry of the Circle
Revision Notes
Active Maths 4 Book 2
Chapter 11
Name:________________________________
Note: Make sure to use the page numbers on the slides to refer back to your Active Maths book to get examples on how to complete the questions.

2. To find the centre and radius
To find the centre and radius. Given the Circle K: x2 + y 2 = r2 (Page 361) K r
Method
Centre: c(0, 0)
Radius = r c

3. Given the centre and radius of a circle, to find the equation of Circle K?
Method
Sub centre & radius into: (x – h)2 + (y – k)2 = r2
If required expand to: x2 + y2 +2gx +2fy + c = 0 r
c(h, k)

4. To find the centre and radius
To find the centre and radius. Given the Circle K: (x – h)2 + (y – k)2 = r2 (Page 363) K r
Method
Centre: c(h, k)
Radius = r c

5. To find centre and radius of K
To find centre and radius of K. Given the circle K: x2 + y2 +2gx +2fy + c = 0? (Page 366) K r
Method
Centre: c(-g, -f)
Radius: c

6. Given equation of circle K, asked a point is on, inside or outside the circle? (Page 367)b
Method
Sub each point into the circle formula K = 0
Answer > outside
Answer = on
Answer < Inside c K

7. Given circle K and the line L to find points of intersection
Given circle K and the line L to find points of intersection? (Page 370) L b
Method
Write the line in terms of y or x.
Sub into the equation of the circle to find the points a K

8. Important to remember TheoremK
Theorem
A line from the centre (c) to the point of tangency (t) is perpendicular to the tangent.
90o c t
radius
Tangent

9. Given equation of Circle K and equation of Tangent T, find the point of intersection?
(Page 370) K T
Method
Write the line in terms of y or x.
Sub into the equation of the circle to find the points t

10. K r a b c To find equation of circle K given end points of diameter?
Method
Centre is midpoint [ab]
Radius is ½|ab| (distance formula)
Sub into circle formula r a b c

11. Given equation of Circle K and asked to find equation of tangent T at given point t?
Method
Find the slope of the radius
Find the perpindicular slope of the line T
Solve the equation of the line using your perpindicular slope and point t c K

12. To find equation of circle K, given that x-axis is tangent to K, and centre c(-f, -g) ?
Method
On x-axis, y = 0 so the point t is (-f, 0)
Find the radius
Sub into circle formula K
c(-g, -f)
r = |f|
X-axis
t(-g, 0)

13. To find equation of circle K, given that y-axis is tangent to K, and centre c(-f, -g) ?
Method
On y-axis, x = 0 so the point t is (0, -g)
Find the radius
Sub into circle formula
r = |g|
t(0, -f)
c(-g, -f) K
y-axis

14. Given equation of Circle K and equation of line L, how do you prove that L is a tangent?
(Page 371) L
Method
Find the distance from the centre of the circle to the line
If the perpendicular distance is equal to the radius then it is a tangent
If the perpendicular distance is not equal to the radius then it is not a tangent K r c

15. Given equation of Circle K and point p, to find equations of tangents from p(x1,y1)?
(Page 374)
Method
Find the centre c and radius r
Sub the point into line formula and let the slope be m giving:
mx – y + (mx1 – y1) = 0
Use the perpindicular distance formula and solve for m:
You will get 2 values for m.
Then sub these 2 values for m back into your line formula to find the equations of the 2 tangents T1 p r K c r T2

16. Given equation of Circle K & Line L: ax + by + c = 0 to find equation of tangents parallel to L?
Method
Find centre c and radius r
Let parallel tangents be:
ax + by + k = 0
Sub into distance from point to line formula and solve: T1 r c K r T2

17. K To prove a locus is a circle? (Page 372) Method c
If the locus of a set of points is a circle it can be written in the form:
x2 + y2 +2gx + 2fy + c = 0
We then can write its centre and radius. r c K

18. Given equations of Circle K and Circle C, to show that they touch internally? (Page 375)
Method
Find distance between centres
If d = r1 - r2 r2 r1 K c2 d c1

19. Given equations of Circle K and Circle C, to show that they touch externally? (Page 375)
Method
Find distance d between centres
If d = r1 + r2 r2 K r1 d c1

20. Given three points and asked to find the equation of the circle containing them?
(Page 376) b
Method
Sub each point into formula:
x2 + y2 + 2gx + 2fy + c = 0
Solve the 3 equations to find: g, f and c,
Sub into circle formula a c

21. Given 2 points on circle and the line L containing the centre, to find the equation of the circle? (Page 377) b
Method
Sub each point into the circle:
x2 + y2 + 2gx + 2fy + c = 0
Sub (-g, -f) into equation of the line
Solve 3 equations to find: g, f and c,
Sub the solutions into circle equation a L

22. Given the equation of a tangent, the point of tangency and one other point on the circle, to find the equation of the circle? (Page 378) L
Method
Sub each point into the circle:
x2 + y2 + 2gx + 2fy + c = 0
Use the tangent & tangent point to find the line L containing the centre.
Sub (-g, -f) into equation of L
Solve 3 equations to find: g, f and c,
Sub solutions into circle equation a b T