Question

$z_1{z}_2\in C,{z_1}^2+{z_2}^2\in R,z_1({z_1}^2-3{z_2}^2)=2$ and $z_2(3{z_1}^2-{z_2}^2)=11$, then the value of ${z_1}^2+{z_2}^2$ is

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

The correct option is D $5$

We have, $z_1({z_1}^2-3{z_2}^2)=2$ ...(1)

and, $z_2(3{z_1}^2-{z_2}^2)=11$ ...(2)

Multiplying (2) by $i$ and subtracting it from (1),w e get

${z_1}^3-3{z_2}^2{z}_1+i(3{z_1}^2{z}_2-{z_2}^3)=2+11i$

$\Rightarrow {(z_1+i{z}_2)}^3=2+11i$ ...(3)

Again, multiplying (2) by $i$ and subtracting it from (1), we get

${z_1}^3-3{z_2}^2{z}_1+i(3{z_1}^2{z}_2-{z_2}^3)=2-11i$

$\Rightarrow {(z_1-i{z}_2)}^3=2-11i$ ...(4)

Multiplying (3) and (4), we get

${({z_1}^2+{z_2}^2)}^3=4+121\Rightarrow {z_1}^2+{z_2}^2=5$