Show that \(\left( \frac { 19+9i }{ 5-3i } \right) ^{ 15 }-\left( \frac { 8+i }{ 1+2i } \right) ^{ 15 }\) is purely imaginary.

Let z₁, z₂ and z₃ be complex numbers such that \(\left| { z }_{ 1 } \right\| =\left| { z }_{ 2 } \right| =\left| { z }_{ 3 } \right| =r>0\) and z₁+ z₂+ z₃ \(\neq \) 0 prove that \(\left| \frac { { z }_{ 1 }{ z }_{ 2 }+{ z }_{ 2 }{ z }_{ 3 }+{ z }_{ 3 }{ z }_{ 1 } }{ { z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 } } \right| \) = r

If z₁, z_2, and z₃ are three complex numbers such that |z₁| = 1, |z₂| = 2|z₃| = 3 and |z₁ + z₂ + z₃| = 1, show that |9z₁z₂ + 4z₁z₂ + z₂z₃| = 6

 If z = x + iy is a complex number such that Im \(\left( \frac { 2z+1 }{ iz+1 } \right) =0\) show that the locus of z is 2x²+ 2y²+ x - 2y = 0

If z = x + iy and arg \(\left( \frac { z-i }{ z+2 } \right) =\frac { \pi }{ 4 } \), then show that x² + y² + 3x - 3y + 2 = 0