1 Co-ordinate Geometry of the Circle - PowerPoint Presentation

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