if( |z-z1|)/(|z-z2|)=3,where z1 and z2 are fixed complex nos. and z is variable the z lies on..
 
1. circle with z1 as its interior pt.
2. circle with z2 as its interior pt.
3. circle with z1 as its exterior pt.
4. circle with z2 as its exterior pt.
5. both 1 and 2

Asked by Sunil Soni | 26th May, 2016, 10:11: AM

Expert Answer:

L e t space Z subscript 1 open parentheses h comma k close parentheses space a n d space Z subscript 2 open parentheses p comma q close parentheses space b e space t w o space f i x e d space c o m p l e x space n u m b e r s.
L e t space Z open parentheses x comma y close parentheses space b e space t h e space v a r i a b l e
N o w space c o n s i d e r space t h e space g i v e n space e q u a t i o n
fraction numerator open vertical bar Z minus Z subscript 1 close vertical bar over denominator open vertical bar Z minus Z subscript 2 close vertical bar end fraction equals 3
rightwards double arrow open square brackets fraction numerator open vertical bar Z minus Z subscript 1 close vertical bar over denominator open vertical bar Z minus Z subscript 2 close vertical bar end fraction close square brackets squared equals 3 squared
rightwards double arrow open vertical bar Z minus Z subscript 1 close vertical bar squared equals 9 open vertical bar Z minus Z subscript 1 close vertical bar squared
rightwards double arrow open square brackets square root of open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared end root close square brackets squared equals 9 open square brackets square root of open parentheses x minus p close parentheses squared plus open parentheses y minus q close parentheses squared end root close square brackets squared
rightwards double arrow open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals 9 open square brackets open parentheses x minus p close parentheses squared plus open parentheses y minus q close parentheses squared close square brackets space space space... left parenthesis 1 right parenthesis
rightwards double arrow 8 x squared plus 8 y squared minus 18 p x plus 2 h x minus 18 q y plus 2 k y plus 9 p squared plus 9 q squared minus h squared minus k squared equals 0 space space space space... left parenthesis 2 right parenthesis
T h e space a b o v e space e q u a t i o n space i s space t h e space e q u a t i o n space o f space t h e space c i r c l e.

N o w space c o n s i d e r space t h e space e x p r e s s i o n
9 open square brackets open parentheses x minus p close parentheses squared plus open parentheses y minus q close parentheses squared close square brackets minus open parentheses x minus h close parentheses squared minus open parentheses y minus k close parentheses squared
N o w space s u b s t i t u t i n g space x equals h space a n d space y equals k comma space w e space h a v e comma
9 open square brackets open parentheses h minus p close parentheses squared plus open parentheses k minus q close parentheses squared close square brackets greater than 0

T h u s space t h e space p o i n t space Z subscript 1 open parentheses h comma k close parentheses space l i e s space o u t space s i d e space t h e space c i r c l e

N o w space c o n s i d e r space t h e space e x p r e s s i o n
9 open square brackets open parentheses x minus p close parentheses squared plus open parentheses y minus q close parentheses squared close square brackets minus open parentheses x minus h close parentheses squared minus open parentheses y minus k close parentheses squared
N o w space s u b s t i t u t i n g space x equals p space a n d space y equals q comma space w e space h a v e comma
minus open parentheses x minus h close parentheses squared minus open parentheses y minus k close parentheses squared less than 0

T h u s space t h e space p o i n t space Z subscript 2 open parentheses p comma q close parentheses space l i e s space i n s i d e space t h e space c i r c l e

Answered by Vimala | 27th May, 2016, 01:05: PM