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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
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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
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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
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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
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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...
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),( 11 yxT
)0,0(O
222 ryxL
y
x1x
1y
LineTangent
gr
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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
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),( 11 yxT
O
y
x
LineTangent
g
),( baA
a1x
1y
b
r
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STEPS EQUATION
Write the general equationof the line
Substitute gradient of the
tangent line g (mg) into m1
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
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222)()( rbyax
2
11 ))(())(( rbybyaxax
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),( 11 yxTy
x
1x
LineTangent
g
),(21
21 BAL
1y
B2
1
A2
1
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STEPS EQUATIONS
Write the general equation
of the line
Substitute gradient of the
tangent line g (mg) into m1
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
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02
1)(
2
11111 CyyBxxAyyxx
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13/10/2012
Nikmah Alfi Nahdiya