110 Circles; their basic properties Ch. 7
Thus as we we may replaceaandbbya/
√
a^2 +b^2 andb/√
a^2 +b^2 , without loss of
generality we may assume thata^2 +b^2 =1. Then the pointZon the line lies on the
circle if(x 0 +bt)^2 +(y 0 −at)^2 −k^2 =0, that is if
t^2 + 2 (bx 0 −ay 0 )t+x^20 +y^20 −k^2 = 0.Ift 1 ,t 2 are the roots of this equation, thent 1 t 2 =x^20 +y^20 −k^2 .AsforRandSwe have
x 1 =x 0 +bt 1 ,y 1 =y 0 −at 1 ,x 2 =x 0 +bt 2 ,y 2 =y 0 −at 2 ,so|P,R|=|t 1 |,|P,S|=|t 2 |. Thus
|P,R||P,S|=|t 1 t 2 |=|x^20 +y^20 −k^2 |,which is constant.
WhenPis exterior to the circle, the roots of the quadratic equation are equal if
(bx 0 −ay 0 )^2 =x^20 +y^20 −k^2 ,and the repeated root is given byt=−(bx 0 −ay 0 ). Then for a point of intersection
T 1 of the line and circle, we have for the coordinates ofT 1
x=x 0 −(bx 0 −ay 0 )b, y=y 0 +(bx 0 −ay 0 )a.Hence
|P,T 1 |^2 =(x−x 0 )^2 +(y−y 0 )^2 =(bx 0 −ay 0 )^2 =x^20 +y^20 −k^2 =|O,P|^2 −k^2.It is also easy to give a synthetic proof as follows. We first takePinterior to the
circle. LetMbe the mid-point ofRandSso thatMis the foot of the perpendicular
fromOtoRS.ThenPis in either[R,M]or[M,S]; we suppose thatP∈[R,M].
S
R
M
P
O
Figure 7.5.
O
R P
T 1
M
S
Then
|P,R||P,S|=(|M,R|−|P,M|)(|M,S|+|P,M|)=|M,R|^2 −|P,M|^2=|M,R|^2 −(
|P,O|^2 −|O,M|^2
)
=
(
|M,R|^2 +|O,M|^2
)
−|P,O|^2
=|O,R|^2 −|P,O|^2 =k^2 −|P,O|^2 ,