Tangent Equation of a Circle PPT 1

7/28/2019 Tangent Equation of a Circle PPT 1

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Determining the equation of a circle and its tangentline.

STANDARD COMPETENCE

Determining the tangent equation of a circle in anysituations.

BASIC COMPETENCE

Determining tangent equation of a circle where the centre

point is O(0,0) and the radius is rthrough a point T(x1,y1)

on a circle. Determining tangent equation of a circle where the centre

point is A(a,b) and the radius is rthrough a point T(x1,y1)

on a circle.

Determining the tangent equation of a circle

x2+y2+Ax+By+C=r2 through a point on the circle T(x1,y

1)

Indicators


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STRAIGHT LINEEQUATION

Gradient of astraight Line

Straight LineEquation

Parallel andPerpendicular Lines

CIRCLEEQUATION

Centre O(0,0) andradius r

Centre A(a,b) andradius r

General Form of aCircle Equation


4/18

Gradient of a Straight Line

Line Equation cmxy

Gradient of the line is m

Line Equation 0 CbyAx

Gradient of the line is A/B

For a line passes through 2 points and),( 11 yxP ),( 22 yxR

Gradient of the line is12

12

xx

yy


5/18

Straight Line Equation

Equation of the line which

passes through point (x1,y1)

and the gradient is m

Equation of the line which

passes through the points

(x1,y1) and (x2,y2)

)( 11 xxmyy

12

1

12

1

xx

xx

yy

yy


6/18

Parallel and Perpendicular Lines

Parallel

Gradient of both

lines are same.

If line g and line h

parallel, then :

Perpendicular

If line g and line h

perpendicular,

then:

hg mm 1. hg mm


7/18

Circle Equation where the centre is O(0,0) and having

radius r.

Circle Equation where the centre is A(a,b) and having

radiusr.

General Form of Circle Equation

222ryx

222 )()( rbyax

022 CByAxyx

CIRCLE EQUATION


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Now,Lets find the tangent equation of the cirlce...


9/18

),( 11 yxT

)0,0(O

222 ryxL

y

x1x

1y

LineTangent

gr


10/18

STEPS EQUATIONS

Write the general equationof the line

Substitute gradient of the

tangent line g (mg) into m1

Multiply both sides by y1

Expand both parts of equation

Collect the variable inquadratic form in right side and

the others in the left side

Remember that x2+y2= r2

)( 111 xxmyy

)( 11

11 xx

y

xyy

)()( 1111 xxxyyy

2

11

2

11 xxxyyy

2

1

2

111 yxyyxx

2

11 ryyxx


11/18


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),( 11 yxT

O

y

x

LineTangent

g

),( baA

a1x

1y

b

r


13/18

STEPS EQUATION

Write the general equationof the line



Multiply both sides by (y1-b)

Expand both sides by

multiply all variables.

Move -x1xand axto the right

side and move -y12 and byto

the left side.

)( 111 xxmyy

)( 11

11 xx

by

axyy

))(()( 1111 xxaxyyby

1

2

111

2

11 axaxxxxbybyyyy

,

,

11

2

1

2

111 byaxyxbyaxyyxx


14/18

222)()( rbyax

2

11 ))(())(( rbybyaxax


15/18

),( 11 yxTy

x

1x

LineTangent

g

),(21

21 BAL

1y

B2

1

A2

1


16/18

STEPS EQUATIONS

Write the general equation

of the line



Multiply both sides by

(y1+b)

Expand both parts of

equations

)( 111 xxmyy

)( 12

1

1

21

1

1 xx

By

Axyy

))(())(( 121

1121

1 xxAxyyBy

1

2

111

2

112

1

2

1

2

1

2

1AxAxxxxByByyyy


17/18

02

1)(

2

11111 CyyBxxAyyxx


18/18

13/10/2012

Nikmah Alfi Nahdiya

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Tangent Equation of a Circle PPT 1