Question

prove geometrically that \[\left| \frac{z-3i}{z-i} \right|=5\] is a circle.

Answer 1

can we put z = x+iy? i guess not

Answer 2

nope..

Answer 3

thats a stupid question

Answer 4

u are not allowed to use any formulae

Answer 5

its a circle...now consider the points (0,3) and (0,1)...find the point (0,y) which divides them in the ratio 5:1 externally...and internally. the external point is the centre and for the radius u hav the internal point for finding it.
understood?

Answer 6

arre all of us kno its a circle man..he has to PROVE it

Answer 7

an that too geometrically

Answer 8

oh! then putting x+iy will help u get it....

Answer 9

i need a proof...not a case

Answer 10

cant put in that..geometrical proof

Answer 11

u cant put x+iy..i need a geometrical proof...u r not allowed to use algebra

Answer 12

oh! gimme sometime please :)

Answer 13

i know how to visualize the other locus which represents the arc of a circle in terms of
arg[(z-z1)/(z-z2)] = theta..but this ones just not striking

Answer 14

lol, there is no such thing as a geometric proof

Answer 15

it needs algebra,

Answer 16

wat exactly u mean by 'geometry' ? wat all r u allowed to use?

Answer 17

if we assume that the thing inside the modulus is a complex number then we can say that mod of that thing gives the distance of the arbitrary point from the centre which is constant and hence the given eqn is a circle. But now we need to prove the assumption that the thing inside the mod is a complex numbers i.e. an arbitrary point in the complex plane "geometrically". any idea with geometrical representation of complex division?

Answer 18

the equation simply means that the ratio of a complex number from 2 fixed pts on the argand plane is 5

Answer 19

ya..but how do u say that the points will describe a circle then? any geometrical properties?

Answer 20

haha..i dunno ..point is we were just told it shows a circle..and told how to find its centre and radius..like brackett told u...no1 ever told us why it was circle

Answer 21

o man! thats not the way complex numbers were created1 lol :P

Answer 22

lol////

Answer 23

if ratio of distances of an arbitrary point from 2 fixed points on the y-axis is a constant shouldnt the locus be a line parallel to x-axis?

Answer 24

weve been taught its a line only when the ratio is 1

Answer 25

wht u r saying is true only when the ratio is 1..then its the perp bisector

Answer 26

for other than 1 its a circle?

Answer 27

yes

Answer 28

i checked my notes..no proof..sorry

Answer 29

ya...used locus concepts..u r true..but how to prove geometrically?

Answer 30

nope....if the ratio given is >1 (here5>1) it represents a circle...

Answer 31

thats what i said