118 Circles; their basic properties Ch. 7
7.6.3 Radicalaxisandcoaxalcircles ....................
In 7.4.2 our proof showed that if a line through the pointZmeets the circleC(Z 1 ,k 1 )
at the pointsRandSthenZRZS=|Z 1 ,Z|^2 −k^21 depends only on the circle and the
pointZ. We call this expression thepowerof the pointZwith respect to this circle.
To cater for degenerate cases, whenk 1 =0 we also call|Z 1 ,Z|^2 the power ofZwith
respect to the pointZ 1.
We letC 1 denote eitherC(Z 1 ,k 1 )orZ 1 and similarly considerC 2 which is either
C(Z 2 ,k 2 )orZ 2. We askfor what points Z its powers with respect toC 1 andC 2 are
equal. This occurs when|Z 1 ,Z|^2 −k^21 =|Z 2 ,Z|^2 −k^22 which simplifies to
2 (x 2 −x 1 )x+ 2 (y 2 −y 1 )y+x^21 +y^21 −k^21 +x^22 +y^22 −k^22 = 0.
IfZ 1 =Z 2 this is the equation of a line which is called theradical axisofC 1 andC 2.
It is always perpendicular to the lineZ 1 Z 2 and it passes through any points whichC 1
andC 2 have in common.
More generally we also askfor what points Z its powers with respect toC 1 and
C 2 have a constant ratio. For a real numberλwhich is not equal to 1 we consider
when
|Z 1 ,Z|^2 −k 12 =λ
[
|Z 2 ,Z|^2 −k^22
]
. (7.6.6)
Whenλ=0 this yieldsC 1 and by consideringμ
[
|Z 1 ,Z|^2 −k 12
]
=|Z 2 ,Z|^2 −k 22 as
well, we also includeC 2.
Now (7.6.6) expands to
x^2 +y^2 − 2
x 1 −λx 2
1 −λ
x− 2
y 1 −λy 2
1 −λ
y+
x^21 +y^21 −k^21 −λ(x^22 +y^22 −k^22 )
1 −λ
= 0 ,
and on completing the squares in bothxandyit becomes
[
x−
x 1 −λx 2
1 −λ
] 2
+
[
y−
y 1 −λy 2
1 −λ
] 2
=
1
( 1 −λ)^2
{
k 12 +
[
(x 1 −x 2 )^2 +(y 1 −y 2 )^2 −k^21 −k^22
]
λ+k^22 λ^2
}
.
This quadratic expression inλis postive when|λ|is large, so it has either a positive
minimum, or its minimum is 0 attained atλ 1 , say, or it has a negative minimum and
so has the value 0 atλ 2 andλ 3 , say, whereλ 2 <λ 3. In the first of these cases (7.6.6)
always represents a circle and in the second case it represents a circle for allλ=λ 1
and a point forλ=λ 1. In the third case it represents a circle when eitherλ<λ 2 or
λ>λ 3 , it represents a point when eitherλ=λ 2 orλ=λ 3 , and it represents an empty
locus whenλ 2 <λ<λ 3. Thus it is the equation of a circle, a point or an empty locus.
Suppose that we consider two of these loci, corresponding to the valuesλ 4 andλ 5
ofλ. They will then have equations
x^2 +y^2 − 2
x 1 −λ 4 x 2
1 −λ 4
x− 2
y 1 −λ 4 y 2
1 −λ 4
y+
x^21 +y^21 −k^21 −λ 4 (x^22 +y^22 −k^22 )
1 −λ 4
= 0 ,
x^2 +y^2 − 2
x 1 −λ 5 x 2
1 −λ 5
x− 2
y 1 −λ 5 y 2
1 −λ 5
y+
x^21 +y^21 −k^21 −λ 5 (x^22 +y^22 −k^22 )
1 −λ 5