Question

If $|z_1|=2,|z_2|=3,|z_3|=4$ and $|2z_1+3z_2+4z_3|=0$, then absolute value of $8z_2{z}_2+27z_3{z}_1+64z_1{z}_2$ must be equal to

Answer 1

Let $z=2z_1+3z_2+4z_3$

$|z{|}^2=z\overline{z}$

$⟹|z{|}^2=4|z_1{|}^2+9|z_2{|}^2+16|z_3{|}^2+6(z_1{\overline{z}}_2+{\overline{z}}_1{z}_2)+12(z_2{\overline{z}}_3+{\overline{z}}_2{z}_3)+8(z_1{\overline{z}}_3+{\overline{z}}_1{z}_3)=0$ ........{$|z|=0$ is given}

$.............................\left(i\right)$

Let $y=8z_2{z}_3+27z_3{z}_1+64z_1{z}_2=2^3{z}_2{z}_3+3^3{z}_3{z}_1+2^6{z}_1{z}_2$

Now, $|y{|}^2=y\overline{y}$

So, $|y{|}^2=2^6|z_2{|}^2|z_3{|}^2+3^6|z_3{|}^2|z_1{|}^2+2^{12}|z_1{|}^2|z_2{|}^2+2^3{3}^3|z_3{|}^2(z_1{\overline{z}}_2+{\overline{z}}_1{z}_2)+3^3{2}^6|z_1{|}^2(z_2{\overline{z}}_3+{\overline{z}}_2{z}_3)$ $+2^9|z_2{|}^2(z_1{\overline{z}}_3+{\overline{z}}_1{z}_3)$

$⟹|y{|}^2=2^6{3}^2{4}^2+3^6{4}^2{2}^2+2^{12}{2}^2{3}^2+2^2{3}^2{4}^2\left[6\right(z_1{\overline{z}}_2+{\overline{z}}_1{z}_2)+12(z_2{\overline{z}}_3+{\overline{z}}_2{z}_3)+8(z_1{\overline{z}}_3+{\overline{z}}_1{z}_3\left)\right]$

$⟹|y{|}^2=2^6{3}^2(2^4+3^4+4^4)-2^2{3}^2{4}^2[4|z_1{|}^2+9|z_2{|}^2+16|z_3{|}^2]$ $........from$ $\left(i\right)$

Here we used the result from the condition of $|z|=0$ and used it to evaluate $|y|$

Substituting values of $|z_1|,|z_2|,|z_3|,$ we get

$|y{|}^2=2^2{3}^2{4}^2\left[\right(2^4+3^4+4^4)-({4.2}^2+{9.3}^2+{16.4}^2\left)\right]$

$⟹|y{|}^2=2^2{3}^2{4}^2[2^4+3^4+4^4-2^4-3^4-4^4]=0$

$⟹|y|=0$

$|8z_2{z}_3+27z_3{z}_1+64z_1{z}_2|=0$