document.write( "Question 16100: z1 and z2 are two complex numbers. if |z1+z2|=|z1|+|z2| then show that arg(z1)=arg(z2) \n" ); document.write( "

Algebra.Com's Answer #7942 by venugopalramana(3286) $\"\"$ $\"About$

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|z1+z2|=|z1|+|z2|
\n" ); document.write( "(|z1+z2|) ^2 = (|z1|+|z2| )^2 = |z1|^2+|z2|^2+2|z1||z2|\r
\n" ); document.write( "\n" ); document.write( "let z1=x1+iy1
\n" ); document.write( " z2=x2+iy2
\n" ); document.write( "hence
\n" ); document.write( " ( |x1+iy1+x2+iy2| )^2 = ( |x1+iy1| )^2+( |x2+iy2| )^2 +2( |x1+iy1| )( |x2+iy2| )
\n" ); document.write( " ( |(x1+x2)+i(y1+y2)| ) ^2 = (x1^2+y1^2) +(x2^2+y2^2)+2 $\"sqrt%28%28x1%5E2%2By1%5E2%29%28x2%5E2%2By2%5E2%29%29\"$
\n" ); document.write( " (x1+x2)^2+(y1+y2)^2 = (x1^2+y1^2) +(x2^2+y2^2)+2 $\"sqrt%28%28x1%5E2%2By1%5E2%29%28x2%5E2%2By2%5E2%29%29\"$
\n" ); document.write( "x1^2+x2^2+2x1*x2+y1^2+y2^2+2y1*y2 = (x1^2+y1^2) +(x2^2+y2^2)+2 $\"sqrt%28%28x1%5E2%2By1%5E2%29%28x2%5E2%2By2%5E2%29%29\"$
\n" ); document.write( "2(x1*x2+y1*y2) = 2 $\"sqrt%28%28x1%5E2%2By1%5E2%29%28x2%5E2%2By2%5E2%29%29\"$
\n" ); document.write( "cancelling 2 on either side and squaring both sides
\n" ); document.write( "(x1x2+y1y2)^2 = (x1^2+y1^2)(x2^2+y2^2)
\n" ); document.write( "(x1^2)(x2^2)+(y1^2)(y2^2) +2x1*x2*y1*y2 = (x1^2)(x2^2)+(y1^2)(y2^2)+(x1^2)(y2^2)+(x2^2)(y1^2)
\n" ); document.write( "(x1^2)(y2^2)+(x2^2)(y1^2) - 2x1*x2*y1*y2 = 0
\n" ); document.write( "(x1y2-x2y1)^2 = 0
\n" ); document.write( "x1y2 - x2y1 = 0
\n" ); document.write( "x1y2 = x2y1
\n" ); document.write( "y1/x1 = y2/x2
\n" ); document.write( "Tan (y1/x1) = Tan (y2/x2)
\n" ); document.write( " Hence argument of z1 = argument of z2
\n" ); document.write( " \n" ); document.write( "

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