Question

State condition under which the product of two complex number is purely imaginary

Answer 1

The sum of two complex numbers is a pure imaginary number when either of the two conditions happen:

The real parts of two complex numbers are both zero.

The real parts of two complex numbers have the same absolute value but have different signs.

In other words, if

z1=a1+ib1

z2=a2+ib2

then given that z1+z2=0+ic

⟹(a1+a2)+i(b1+b2)=0+ic

⟹(a1+a2)=0

⟹a1=−a2  or  a1=a2=0