if |z2+iz1|= |z1| +|z2| and |z1|=3 and |z2|=4 then area of tri ABC if affixes of A,B,C are z1 , z2,
and (z2-iz1)/(1-i) resp is...

Asked by Sunil Soni | 26th May, 2016, 09:57: AM

Expert Answer:

open vertical bar z subscript 2 plus i z subscript 1 close vertical bar equals open vertical bar z subscript 1 close vertical bar plus open vertical bar z subscript 2 close vertical bar
rightwards double arrow open vertical bar z subscript 2 plus i z subscript 1 close vertical bar equals open vertical bar z subscript 2 close vertical bar plus open vertical bar i z subscript 1 close vertical bar space left parenthesis sin c e space open vertical bar z subscript 1 close vertical bar equals open vertical bar i z subscript 1 close vertical bar right parenthesis
rightwards double arrow z subscript 2 space a n d space i z subscript 1 space h a s space s a m e space a r g u m e n t space i n space c o m p l e x space p l a n e.
rightwards double arrow z subscript 2 equals k i z subscript 1
T a k i n g space m o d u l u s space o f space b o t h space s i d e s
open vertical bar z subscript 2 close vertical bar equals open vertical bar k i z subscript 1 close vertical bar
rightwards double arrow open vertical bar z subscript 2 close vertical bar equals k open vertical bar z subscript 1 close vertical bar
rightwards double arrow 4 equals k cross times 3
rightwards double arrow k equals 3 over 4
T h e space t h i r d space v e r t e x space o f space t h e space t r a i n g l e space i s space a t space fraction numerator z subscript 2 minus i z subscript 1 over denominator 1 minus i end fraction equals fraction numerator k i z subscript 1 minus i z subscript 1 over denominator 1 minus i end fraction equals fraction numerator i z subscript 1 open parentheses k minus 1 close parentheses over denominator 1 minus i end fraction
A r e a space o f space t r i a n g l e
equals i over 4 open vertical bar table row 1 1 1 row cell z subscript 1 end cell cell z subscript 2 end cell cell z subscript 3 end cell row cell stack z subscript 1 with bar on top end cell cell stack z subscript 2 with bar on top end cell cell stack z subscript 3 with bar on top end cell end table close vertical bar equals i over 4 open vertical bar table row 1 1 1 row cell z subscript 1 end cell cell k i z subscript 1 end cell cell fraction numerator i z subscript 1 open parentheses k minus 1 close parentheses over denominator 1 minus i end fraction end cell row cell top enclose z subscript 1 end enclose end cell cell top enclose k i z subscript 1 end enclose end cell cell top enclose fraction numerator i z subscript 1 open parentheses k minus 1 close parentheses over denominator 1 minus i end fraction end enclose end cell end table close vertical bar
equals i over 4 cross times z subscript 1 top enclose z subscript 1 end enclose open vertical bar table row 1 1 1 row 1 cell k i end cell cell fraction numerator i open parentheses k minus 1 close parentheses over denominator 1 minus i end fraction end cell row 1 cell negative k i end cell cell negative i open parentheses k minus 1 close parentheses fraction numerator 1 over denominator 1 plus i end fraction end cell end table close vertical bar
equals i over 4 cross times open vertical bar z subscript 1 close vertical bar squared open square brackets fraction numerator k open parentheses k minus 1 close parentheses over denominator 1 plus i end fraction minus fraction numerator k open parentheses k minus 1 close parentheses over denominator 1 minus i end fraction plus i open parentheses k minus 1 close parentheses open square brackets fraction numerator 1 over denominator 1 minus i end fraction plus fraction numerator 1 over denominator 1 plus i end fraction close square brackets minus 2 k i close square brackets
equals i over 4 cross times 9 open square brackets fraction numerator negative 2 i k open parentheses k minus 1 close parentheses over denominator 2 end fraction plus i open parentheses k minus 1 close parentheses minus 2 k i close square brackets
equals i over 4 cross times 9 open square brackets negative i k squared plus i k plus i k minus i minus 2 i k close square brackets
equals i over 4 cross times negative 9 i open parentheses k squared plus 1 close parentheses
equals 9 over 4 cross times 25 over 16 equals 225 over 64 space left parenthesis w h e r e space k equals 3 over 4 right parenthesis

Answered by satyajit samal | 30th May, 2016, 12:09: AM