If `z_1, z_2 in C ,z_1^2+z_2^2 in R ,z_1(z_1^2-3z_2^2)=2 and z_2(3z_1^2-z_2^2)=11` , then the value of `z_1 ^2+z_2 ^2` is `10` b. `12` c. `5` d. `8`

Explanation

`z_1(z_1^2 + 3z_2^2) = 2` (1)

`z_2(3z_1^2 - z_2^2) = 11` (2)

multiplying eqn (2) with i and adding (1) and (20

`z_1^3- 3z_1z_2^2 + i(3z_1^2z_2 - z_2^3)= 2+ 11i`

now, `(z_1+ iz_2)^3= 2+ 11i` (A)

now we will`(1) - i(2):`

`z_1^3 - 3z_1z_2^2 - i(3z_1^2z_2 - z_2^3) = 2- 11i`

`(z_1-iz_2)^3 = 2-11i` (B)

...