Sec. 7.6 Sensed distances 119
On subtracting the second of these from the first, and simplifying, we find that their
radical axis is the line with equation
λ 4 −λ 5
( 1 −λ 4 )( 1 −λ 5 )[
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.
As we can cancel the initial fraction we see that these loci have the same radical axis
as did the original pairC 1 andC 2. For this reason all the loci considered are said to
becoaxal.
Exercises
7.1 Prove that a circle cannot have more than one centre. [Hint. IfOandO 1 are
both centres, consider the intersection ofOO 1 with the circle.]7.2 Give an alternative proof of 7.2.1(iv) by showing that if(x−k)^2 +y^2 =k^2 ,
wherek>0, thenx≥ 0.7.3 Prove that if the pointXis interior to the circle(O;k),lis a line containingX,
andM=πl(X),thenMis also an interior point of this circle. Deduce thatlis
a secant line. Show too that ifYis also interior to this circle, then every point
of the segment[X,Y]is also interior.7.4 Show that ifA,B,Care non-collinear points, there is a unique circle to which
the side-linesBC,CA,ABare all tangents.7.5 LetZ 1 ≡(x 1 , 0 ),Z 2 ≡(x 2 , 0 )andZ 3 ≡(x 3 , 0 )be distinct fixed collinear points
andZ 3 not the mid-point ofZ 1 andZ 2 .ForW∈Z 1 Z 2 letlbe the mid-line of
|Z 2 WZ 1. Find the locus ofWsuch that eitherl, or the line throughWperpen-
dicular tol, passes throughZ 3.7.6 Show that the locus of mid-points of chords of a circle on parallel lines is a
diameter.7.7 Show that if two tangents to a circle are parallel, then their points of contact
are at the end-points of a diameter.7.8 Show that if each of the side-lines of a rectangle is a tangent to a given circle,
then it must be a square.7.9 Consider the circleC(O;a)and pointZ 1 ≡(x 1 , 0 )wherex 1 >a>0, so thatZ 1
is an exterior point which lies on the diametral lineAB,whereA≡(a, 0 ),B≡
(−a, 0 ). Show that for all pointsZ≡(x,y)on the circle,|Z 1 ,A|≤|Z 1 ,Z|≤|Z 1 ,B|.