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If z1, z2 in C ,z1^ 2 in R ,z1(z1^ 2-3...

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

Text Solution

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`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)
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