Geometry with Trigonometry

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

= 0.