i. Show that the centers of the circles of an elliptic family (i.e. a set of cir
ID: 1941455 • Letter: I
Question
i. Show that the centers of the circles of an elliptic family (i.e. a set of circles with a commond chord AB) lie on the same straight line. (As shown, this line is the radical axis of the hyperbolic family of the circles orthogonal to all the circles of the elliptic family).
ii. C1, C2, C3 are three circles such that their centers do not lie on the same straight line. let L1, L2, L3 be the radical axes of the pairs C2C3, C3C1, and C1C2 respectively. Show that L1, L2 and L3 are concurrent. What can you say
a) if the centers of C1, C2, C3 are on the same straight line
b) if the two centers coincide
Explanation / Answer
i. Show that the centers of the circles of an elliptic family (i.e. a set of circles with a commond chord AB) lie on the same straight line. (As shown, this line is the radical axis of the hyperbolic family of the circles orthogonal to all the circles of the elliptic family).
ii. C1, C2, C3 are three circles such that their centers do not lie on the same straight line. let L1, L2, L3 be the radical axes of the pairs C2C3, C3C1, and C1C2 respectively. Show that L1, L2 and L3 are concurrent. What can you say
a) if the centers of C1, C2, C3 are on the same straight line
b) if the two centers coincide
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i. Show that the centers of the circles of an elliptic family (i.e. a set of circles with a commond chord AB) lie on the same straight line. (As shown, this line is the radical axis of the hyperbolic family of the circles orthogonal to all the circles of the elliptic family).
LET THE CIRCLES..C1,C2,C3,....ETC... BE REPRESENTED BY THE EQNS.
S'=0.........................................1
S''=0....................................2
S'''=0..ETC..............................3
[WHERE S IS THE STD. FORM OF EQN. OF A CIRCLE ...VIZ...
X^2+Y^2+2GX+2FY+C=0....ETC....]
LET THE COMMON CHORD BE PQ WITH END POINTS OF P[X1,Y1]
AND Q[X2,Y2]
SINCE P & Q LIE ON ALL THE CIRCLES WE HAVE
S'[1]=0=S'[2].......................................4
S''[1]=0=S''[2]..................................5
S'''[1]=0=S'''[2]....................................6
CONSIDE THE EQN.1-EQN.2
S'-S''=0=X^2+Y^2+2G'X+2F'Y+C'-X^2+Y^2+2G''X+2F''Y+C''=0
=X[2G'-2G'']+Y[2F'-2F'']+[C'-C'']=0...THIS IS A LINEAR EQN ..SO IT IS A STRAIGHT LINE
AS PER EQNS. 4 AND 5 , IT IS SATISFIED BY BOTH P AND Q .
HENCE ITS THE EQN . OF THE COMMON CHORD OF THE 2 CIRCLES..C1 AND C2.
HENCE C1C2 IS PERPENDICULAR BISECTOR OF PQ.
SIMILARLY
C2C3 AND C3C1...ETC... ARE PERPENDICULAR BISECTORS OF PQ
HENCE C1,C2,C3...ETC... ARE CONCURRENT ....PROVED....
ii. C1, C2, C3 are three circles such that their centers do not lie on the same straight line. let L1, L2, L3 be the radical axes of the pairs C2C3, C3C1, and C1C2 respectively. Show that L1, L2 and L3 are concurrent.
USING THE SAME NOMEN CLATURE AS ABOVE
L3 THE RADICAL AXIS OF C1 AND C2 IS GIVEN BY EQN.
S'-S''=0..................................7
L1 THE RADICAL AXIS OF C2 AND C3 IS GIVEN BY EQN.
S''-S'''=0.................................8
L2 THE RADICAL AXIS OF C3 AND C1 IS GIVEN BY EQN.
S'''-S'=0..................................9
LET THE INTERSECTION OF L2 AND L3 BE P[H,K]
HENCE WE HAVE
S''[P]-S'''[P]=0.......................10
S'''[P]-S'[P]=0..............................11
ADDING EQNS.10+EQN.11 WE GET
S''[P]-S'[P]=0
THAT IS P SATISFIES L1
HENCE P LIES ON L1
THAT IS P LIES ON L1,L2,L3
HENCE L1,L2,L3 ARE CONCURRENT ..
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What can you say
a) if the centers of C1, C2, C3 are on the same straight line....
AS SHOWN ABOVE THEY HAVE ONE COMMON RADICAL AXIS WHICH IS
THE COMMON CHORD OF ALL OF THEM.
b) if the two centers coincide
WE HAVE 2 CONCENTRIC CIRCLES
THEN THEY DO NOT HAVE ANY REAL RADICAL AXIS.
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