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 ID: 618073
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Slide1
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. Slide2
To find the centre and radius. Given the Circle K:
x2 + y 2 = r2 (Page 361)
MethodCentre: c(0, 0)Radius = r
K
r
c
2Slide3
Given the centre and radius of a circle, to find the equation of Circle K
?
K
r
Method
Sub centre & radius into:
(x – h)2
+ (y – k)2 = r2
If required expand to: x
2 + y2 +2gx +2fy + c = 0
c(h, k)
3Slide4
To find the centre and radius. Given the Circle K:
(x – h)2 + (y – k)2 = r2 (Page 363)
MethodCentre: c(h, k)Radius = r
K
r
c
4Slide5
To find centre and radius of K. G
iven the
circle K: x2 + y2 +2gx +2fy + c = 0? (Page 366)
KMethod
Centre: c(-g, -f)Radius:
r
c
5Slide6
Given equation of circle K, asked
a point
is on, inside or outside the circle
? (Page 367)
a
Method
Sub each point into the circle formula K = 0
Answer > 0 outside
Answer = 0 on
Answer < 0
Inside
b
c
K
6Slide7
Given circle K and the line L to find points of intersection
? (Page 370)
a
b
L
K
7
Method
Write the line in terms of y or x.
Sub into the equation of the circle to find the pointsSlide8
Important to remember
Theorem
A line from the centre (c) to the point of tangency (t) is perpendicular to the tangent.
c
8
90
o
Tangent
K
radius
tSlide9
Given equation of Circle K and equation of Tangent T, find the point of intersection
?
(Page 370)
K
T
9
t
MethodWrite the line in terms of y or x.
Sub into the equation of the circle to find the pointsSlide10
To find equation of circle K given end points of diameter?
K
Method
Centre is midpoint [ab]Radius is ½|ab| (distance formula)
Sub into circle formula
a
10
b
c
rSlide11
Given equation of Circle K and asked to find equation of tangent T at given point t?
K
t
Method
Find the slope of the radius
Find the
perpindicular
slope of
the line TSolve the equation of the line using your
perpindicular slope and point
c
T
11Slide12
To find equation of circle K, given that x-axis is tangent to K, and centre c(-f, -g) ?
X-axis
Method
On x-axis, y = 0 so the point t is (-f, 0)Find the radius
Sub into circle formula
c(-g, -f)
K
12
t(-g, 0)
r = |f|Slide13
To find equation of circle K, given that y-axis is tangent to K, and centre c(-f, -g) ?
y-axis
Method
On y-axis, x = 0 so the point t is (0, -g)Find the radius
Sub into circle formula
c(-g, -f)
K
13
t(0, -f)
r = |g|Slide14
Given equation of Circle K and equation of line L, how do you prove that L is a tangent
?
(Page 371)
K
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 tangentIf the perpendicular distance is not equal to the radius then it is
not
a tangent
14
r
cSlide15
Given equation of Circle K and point p, to find equations of tangents from p(x
1
,y1)? (Page 374)K
c
p
T
1
r
15
T
2
r
Method
Find
the centre
c and radius r
Sub
the point into
line formula
and let the slope be m giving
:
mx
– y + (mx
1
– y
1
) = 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 Slide16
Given equation of Circle K & Line L:
ax + by + c = 0
to find equation of tangents parallel to L?
K
r
Method
Find centre c and radius rLet parallel tangents be:
ax + by + k = 0Sub into distance from point to line formula and solve:
c
L
16
T
1
T
2
rSlide17
To prove a locus is a circle
? (Page 372)
Method
If the locus of a set of points is a circle it can be written in the form: x2 + y2 +2gx + 2fy + c = 0We then can write its centre and radius.
c
K
17
rSlide18
Given equations of Circle K and Circle C, to show that they touch internally
? (Page 375)
K
MethodFind distance between centresIf
d = r1 - r
2
C
r
1
r
2
d
18
c
1
c
2Slide19
Given equations of Circle K and Circle C, to show that they touch externally
? (Page 375)
K
MethodFind distance d between centresIf
d = r1 + r
2
C
r
1
r
2
d
19
c
1
c
2Slide20
Given three points and asked to find the equation of the circle containing them
?
(Page 376)
a
Method
Sub each point into formula:
x2 + y2
+ 2gx + 2fy + c = 0Solve the 3 equations to find: g, f and c,
Sub into circle formula
b
c
20Slide21
Given 2 points on circle and the line L containing the centre, to find the equation of the circle
? (Page 377)
a
Method
Sub each point into the circle:
x
2 + y2
+ 2gx + 2fy + c = 0Sub (-g, -f) into equation of the line
Solve 3 equations to find: g, f and c, Sub the solutions into circle equation
b
L
21Slide22
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)
a
Method
Sub each point into the circle:
x2 + y
2 + 2gx + 2fy + c = 0Use the tangent & tangent point to find the line L containing the centre.
Sub (-g, -f) into equation of LSolve 3 equations to find: g, f and c, Sub solutions into circle equation
b
T
22
L