Prove that `|(z_(1))/(z_(2))|=(|z_(1)|)/(|z_(2)|)`

If Z_(1)=1+i and Z_(2)=2+2i , then which of the following is not true. (A) |z_(1)z_(2)|=|z_(1)||z_(2)| (B) |z_(1)+z_(2)|=|z_(1)|+|z_(2)| (C) |z_(1)-z_(2)|=|z_(1)|-|z_(2)| (D) |(z_(1))/(z_(2))|=(|z_(1)|)/(|z_(2)|)

Prove that |z_(1)+z_(2)|^(2)=|z_(1)|^(2)+|z_(2)|^(2),quad if z_(1)/z_(2) is purely imaginary.

If z_(1) and z_(2)(ne0) are two complex numbers, prove that: (i) |z_(1)z_(2)|=|z_(1)||z_(2)| (ii) |(z_(1))/(z_(2))|=(|z_(1)|)/(|z_(2)|),z_(2)ne0 .

For any two complex number z_(1) and z_(2) prove that: |z_(1)+z_(2)|>=|z_(1)|-|z_(2)|

For any two complex number z_(1) and z_(2) prove that: |z_(1)-z_(2)|>=|z_(1)|-|z_(2)|

For any two complex number z_(1) and z_(2) prove that: |z_(1)+z_(2)|<=|z_(1)|+|z_(2)|

For any two complex number z_(1) and z_(2) prove that: |z_(1)-z_(2)|<=|z_(1)|+|z_(2)|

For any two complex numbers z_(1) and z_(2), prove that |z_(1)+z_(2)| =|z_(1)|-|z_(2)| and |z_(1)-z_(2)|>=|z_(1)|-|z_(2)|

Consider an ellipse having its foci at A(z_(1)) and B(z_(2)) in the Argand plane.If the eccentricity of the ellipse be e an it is known that origin is an interior point of the ellipse, then prove that e in(0,(|z_(1)-z_(2)|)/(|z_(1)|+|z_(2)|))

If z_(1) and z_(2) are two complex number such that |z_(1)|<1<|z_(2)| then prove that |(1-z_(1)bar(z)_(2))/(z_(1)-z_(2))|<1