IIT JEE , AIEEE Forum - Mathematics , Algebra

jaysunantony

If z, x and w are complex no.s in the Argand plane, then find locus of x:

|x-z| = k |x-w| ,k belongs to positive Real no. System, k not eql to 1.

what the eqn represent.......???.....Pls gve exp.......Thanks...

gcch29

the locus of x is circle
put x =h +ir
z =a+ib
w=c+id
and solve simply

catch_arnnie

i also think the ans. is circle

see, |x-z| gives the distance of x from z & |z-w| gives the distance of x from w. since, |x-z|/|x-w| = k where k is not equal to 1, so this would form a circle....

jaysunantony

BUT.........

pls explain...............

for a circle, SP/PM OR ECCENTRICITY IS zero.........

and now, pls explain...........sure rates for explaination........

pls...pls........

lazycol

its a circle;let x b pt A,z b B,w b C

such circles r called appolonius circle

ie if in 3angle ABC, AB : AC = c:d {here c/d =k} den locus A is a circle calld appolonius

try 2 draw a fig by urself

PROOF:

find pt P & Q such dat B,P,C & B,C,Q r collinear (P bw B&C , Q not in bw)

BP / PC = BQ/CQ = C/D = AB/AC

in other words ,divide BC internally & externally ay P & Q in d ratio c/b

AP & AQ r bisectors of interior & exterior angles at A

so anglr PAQ = 90 degrees

A always lies on circle described on PQ as diameter

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