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if |z1+z2|=|z1|+|z2|, then prove that a ...

if `|z_1+z_2|=|z_1|+|z_2|,` then prove that `a r g(z_1)=a r g(z_2)` if `|z_1-z_2|=|z_1|+|z_2|,` then prove that `a r g(z_1)=a r g(z_2)=pi`

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To prove the given statements, we will analyze each case step by step. ### Case 1: Prove that if \( |z_1 + z_2| = |z_1| + |z_2| \), then \( \arg(z_1) = \arg(z_2) \). 1. **Start with the given equation**: \[ |z_1 + z_2| = |z_1| + |z_2| \] ...
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CENGAGE ENGLISH-COMPLEX NUMBERS-ILLUSTRATION
  1. If z1a n dz2 are two complex numbers and c >0 , then prove that |z1+z2...

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  2. If z1, z2, z3, z4 are the affixes of four point in the Argand plane, z...

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  3. if |z1+z2|=|z1|+|z2|, then prove that a r g(z1)=a r g(z2) if |z1-z2|=...

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  4. Show that the area of the triangle on the Argand diagram formed by the...

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  5. Find the minimum value of |z-1| if ||z-3|-|z+1||=2.

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  6. Find the greatest and the least value of |z1+z2| ifz1=24+7ia n d|z2|=6...

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  7. If z is a complex number, then find the minimum value of |z|+|z-1|+|2z...

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  8. If |z1-1|lt=,|z2-2|lt=2,|z(33)|lt=3, then find the greatest value of |...

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  9. Prove that following inequalities: (i) |(z)/(|z|) -1| le |arg z| (...

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  10. Identify the locus of z if z = a +(r^2)/(z-a),> 0.

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  11. If z is any complex number such that |3z-2|+|3z+2|=4 , then identify t...

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  12. If |z|=1 and let omega=((1-z)^2)/(1-z^2) , then prove that the locus o...

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  13. Let z be a complex number having the argument theta ,0 < theta < pi/2,...

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  14. How many solutions the system of equations ||z + 4 |-|z-3i|| =5 and...

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  15. Prove that |Z-Z1|^2+|Z-Z2|^2=a will represent a real circle [with cent...

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  16. If |z-2-3i|^(2) + |z- 5 - 7i|^(2)= lambda respresents the equation ...

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  17. If (|2z - 3|)/(|z-i|)= k is the equation of circle with complex numbe...

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  18. Find the point of intersection of the curves a r g(z-3i)=(3pi)/4a n d ...

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  19. If complex numbers z(1)z(2) and z(3) are such that |z(1)| = |z(2)| = |...

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  20. If the triangle fromed by complex numbers z(1), z(2) and z(3) is eq...

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