Co-Ordinate Geometry of the Circle - PowerPoint Presentation

Slide1

Co-Ordinate Geometry of the Circle - Outcomes

Recognise the equation of a circle.Solve problems about circles centred at the origin.Solve problems about circles not centred at the origin.Determine whether a given point is inside or outside a circle.Solve problems about the intersection of lines and circles.Solve problems about tangents.Solve problems about intersecting circles.

1Slide2

Recognise the Equation of a Circle

Recall that the equation of a line contains an and a .e.g.

is a line

e.g.

is a lineEquations of circles have at least an and a . They may also have an and a but these are not required.e.g. is a circlee.g. is a circlee.g. is a circle

2Slide3

Solve Problems about Circles Centred at

Circles are the locus of points which are one radius away from their centre.

3Slide4

Solve Problems about Circles Centred at

For a circle centred at the origin

, the equation is:

Formula: , where is the radius of the circle.e.g. for radius 3: 4What other formula has two squares adding to equal another square?Slide5

Solve Problems about Circles Centred at

Write down the equations of the circles centred at the origin with each of the following radiuses:

radius = 3

radius = 4radius = 5radius = 6radius = radius = radius = 25 5

Write down the radiuses from the following circle equations:

 Slide6

Solve Problems about Circles Centred at

The equation of a circle describes every point on the

circumference

.We can use the equation to find out if a point is on the circle or to figure out the radius.e.g. Which of the following circles is the point on?

6

Just like when you did this with lines!Slide7

Solve Problems about Circles Centred at

e.g. Find the radius and equation of the circles centred at the origin passing through the points:

 7

 Slide8

Solve Problems about Circles Centred at

The circle

has equation

.Verify that is on the circle .Write down the co-ordinates of a point that lies outside and give a reason for your answer. 82003 OL P2 Q3

The circle

has equation

.

Write down the radius of

.

The radius of another circle is twice the radius of

. The centre of this circle is

. Write down its equation.

A circle has equation

. The points

and

are on the circle. Verify that

is a diameter of the circle.

2004 OL P2 Q3Slide9

Solve Problems about Circles Centred at

A circle

has centre

and diameter 8 units.Show on a co-ordinate diagram.Find the equation of .Prove that the point is inside and that the point is outside it. 92011 OL P2 Q2Slide10

SP about Circles not Centred at

The centres and equations shown are:

10

CentreEquation

Centre

Equation

What is the equation of a circle with centre

and radius

?

 Slide11

SP about Circles not Centred at

The equation of a circle with centre

and radius

is:Formula: e.g. The circle has centre and radius . Its equation is therefore:

11

This is one of the few situations where

not

distributing is considered “simpler”Slide12

SP about Circles not Centred at

Write down the equations of the circles with the following centres and radiuses:

12

QuestionCentreRadiusa)2b)4c)3d)7e)1f)5

Question

Centre

Radius

a)

2

b)

4

c)

3

d)

7

e)

1

f)

5Slide13

SP about Circles not Centred at

Write down the centre and radius of each of the following circles:

13Slide14

SP about Circles not Centred at

Find the equations of the circles with given centres and passing through the given point:

14

QuestionCentrePassing througha)b)c)d)e)

f)

Question

Centre

Passing through

a)

b)

c)

d)

e)

f)Slide15

SP about Circles not Centred at

is a circle with centre

. It passes through the point

.Find the equation of .The point is on the circle . Find the two possible values of . 152004 OL P2 Q3

The circle

has equation

.

Write down the co-ordinates of the centre of

.

The point

is one end-point of a diameter of

. Find the co-ordinates of the other end-point.

The point

is on the circle

. Find the two values of

.

2005 OL P2 Q3Slide16

SP about Circles not Centred at

The vertices of a right-angled triangle are

,

and . The circle passes through the points , and .On a co-ordinate diagram, draw the triangle . Mark the point , the centre of , and draw .Find the equation of .Find the equation of , the image of under the translation . 16Recall corollary 3: each angle in a semicircle is a right angle

2006 OL P2 Q3Slide17

SP about Circles not Centred at

Draw the circle

. Show your scale on both axes.

Verify, using algebra, that is on .Find the equation of the circle with centre that passes through the point . 172015 OL P2 Q3Slide18

Determine if a Point is Inside or Outside

How do the points shown compare to the radius?

18Slide19

Determine if a Point is Inside or Outside

The equation of a circle is

.

A point on the circle will fit this equation.Any other point will be either too big or too small.For a point , the following applies: 19EquationRelationship to Circle

Inside

On

Outside

Equation

Relationship to Circle

Inside

On

Outside

This works because the equation of a circle is essentially

Pythagoras’ Theorem

, which is also essentially the distance formula.Slide20

Determine if a Point is Inside or Outside

Given a circle

, determine whether the following points are inside, outside, or on

.  20Slide21

Determine if a Point is Inside or Outside

Given a circle

, determine whether the following points are inside, outside, or on

.  

21Slide22

SP about Intersection of Lines and Circles

Recall from co-ordinate geometry of the line, we can tell whether shapes intersect by:Graphing them and seeing visually if / where they intersect, orSolving their equations simultaneously and algebraically determining common solutions (i.e. intersections).e.g. Determine the intersection points of

and

by drawing a graph.

Verify your answer by finding the intersection points algebraically. 22Slide23

SP about Intersection of Lines and Circles

To graph the line and circle, we need points.For a circle, its centre and radius will allow us to draw it accurately. (centre , radius

)

For a line (

), substitute values as before: 23

 Slide24

SP about Intersection of Lines and Circles

24Slide25

SP about Intersection of Lines and Circles

To verify the answer algebraically, we require simultaneous equations.Elimination will not work for line-circle intersections.Substitution or transitivity are the only methods that work.

25Slide26

SP about Intersection of Lines and Circles

Rearrange the line equation to or

.

Substitute this into the circle equation.

 26Slide27

SP about Intersection of Lines and Circles

Work through the algebra.Solve the quadratic at the end to get either or

.

27Slide28

SP about Intersection of Lines and Circles

Substitute back into the line equation.Write down the resulting points.28Slide29

SP about Intersection of Lines and Circles

The line

intersects the circle

at the points

and .Draw a co-ordinate diagram on graph paper showing the line, circle and the points of intersection.Verify your answer by finding the points of intersection algebraically. 292003 OL P2 Q3

The line

intersects the circle

at the points

and

.

Draw a co-ordinate diagram on graph paper showing the line, circle and the points of intersection.

Verify your answer by finding the points of intersection algebraically.

2005 OL P2 Q3Slide30

SP about Intersection of Lines and Circles

The line

intersects the circle

. Find the co-ordinates of the points of intersection by drawing a graph and verify your answers by finding the intersection points algebraically.

 302004 HL P2 Q3 [edited]Slide31

Solve Problems about Tangents

A tangent to a circle touches the circle at exactly one point. This is called the point of contact of the tangent.Theorem 20: Each tangent is perpendicular to the radius that goes to that point of contact.

31Slide32

Solve Problems about Tangents

e.g. Find the equation of the tangent to the circle

at the point

.

To get an equation of a line, we need a point and a slope. We have a point. The slope is perpendicular to the radius at the point of contact:centre: ; point of contact  32

 Slide33

Solve Problems about Tangents

The circle has equation

The line

is tangent to the circle

at the point Verify that the point is on .Find the slope of .Find the equation of .The line is another tangent to and is parallel to . Find the co-ordinates of the point at which touches . 332006 OL P2 Q3Hint: if two tangents to the same circle are parallel, they must be at opposite ends of a diameter.Slide34

Solve Problems about Tangents

A tangent is drawn to the circle

at the point

. This tangent crosses the

-axis at . Find the value of . 342008 HL P2 Q1

The circle

has equation

. Write down the co-ordinates of

, the centre of

, and

, the radius of

.

Show that the point

is on the circle

.

Find the slope of the radius

.

Hence, find the equation of

, the tangent to

at

.

A second line

is tangent to

at the point and . Find the co-ordinates of . 2014 OL P2 Q3Slide35

Solve Problems about Intersecting Circles

If two circles touch at a single point, their centres and the point of contact are collinear (on the same line).Circles may touch internally or externally and the distance between their centres,

depends on their radiuses:

35Slide36

Solve Problems about Intersecting Circles

36

Internally:

Externally:  Slide37

Solve Problems about Intersecting Circles

e.g. Two circles,

and

intersect at a single point

.Do the circles touch internally or externally?If , find the equation of their common tangent.,

,

Internally:

Externally:

So they touch externally.

37

Just use a calculator for thisSlide38

Solve Problems about Intersecting Circles

Need a point and a slope. is a point on the tangent. Slope is perpendicular to the radiuses.

38

 Slide39

Solve Problems about Intersecting Circles

The diagram shows two circles

and

of equal radius.

has centre and it cuts the -axis at .Find the equation of .Show that the point is on .The two circles touch at . is on the line joining the two centres. Find the equation of .Find the equation of the common tangent at . 392012 OL P2 Q4Slide40

Solve Problems about Intersecting Circles

A circle has centre

and diameter 8 units.

Show

on a co-ordinate diagram.Find the equation of .Prove that the point is inside and that the point is outside it.Another circle, has centre and just touches the circle . Show on your diagram in part (a) and find the equation of . 402011 OL P2 Q2