Let z abi be a complex number where a and b are real If 4z

Let z = a+bi be a complex number, where a and b are real. If 4|z| = 5z 15i, evaluate a and b.

4|z| = 5z 15i

z = a +ib ; |z| = sqrt(a^2 +b^2)

4sqrt(a^2 +b^2) = 5( a+ib) -15i

4sqrt(a^2 +b^2) = 5a +i(5b -15)

4sqrt(a^2 +b^2) + i*0 = 5a + i(5b -15)

Compare real and imaginary parts on both sides:

4sqrt(a^2 +b^2) = 5a ------ (1)

5b -15 =0 ----> b = 3

Substituting b =3 in equation (1)

4sqrt(a^2 + 9) = 5a

Squaring both sides: 16(a^2 +9) = 25a^2

a^2 +9 = 25a^2/16

a^2 ( 1- 25/16) = -9

a^2( -9/16) = -9

a^2 = 16

a = + / - 4

z = a+ bi = 4 + 3i ; -4 +3i