Question

Let $z_r(1\le r\le 4)$ be complex numbers such that $|z_r|=\sqrt{r+1}and|30z_1+20z_2+15z_3+12z_4|=k|z_1{z}_2{z}_3+z_2{z}_3{z}_4+z_3{z}_4{z}_1+z_4{z}_1{z}_2|$. Then value of $k$ equals ?

• A. $|z_1{z}_2{z}_3|$
• B. $|z_2{z}_3{z}_4|$
• C. $|z_3{z}_4{z}_1|$
• D. $|z_4{z}_1{z}_2|$

Answer 1

The correct option is D $|z_4{z}_1{z}_2|$

$|z_1|=\sqrt{2},|z_2|=\sqrt{3},|z_3|=\sqrt{4},|z_4|=\sqrt{5}$

$|30z_1+20z_2+15z_3+12z_4|=k|z_1{z}_2{z}_3+z_2{z}_3{z}_4+z_3{z}_4{z}_1+z_4{z}_1{z}_2|$

$\overline{|30z_1+20z_2+15z_3+12z_4|}=k|z_1{z}_2{z}_3+z_2{z}_3{z}_4+z_3{z}_4{z}_1+z_4{z}_1{z}_2|$

$|30\overline{z_1}+20\overline{z_2}+15\overline{z_3}+12\overline{z_4}|=k|z_1{z}_2{z}_3+z_2{z}_3{z}_4+z_3{z}_4{z}_1+z_4{z}_1{z}_2$

$z_1\overline{z_1}=2,z_2\overline{z_2}=3,z_3\overline{z_3}=4,z_4\overline{z_4}=5$

$60|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}+\frac{1}{z_4}|=k|z_1{z}_2{z}_3+z_2{z}_3{z}_4+z_3{z}_4{z}_1+z_4{z}_1{z}_2|$

$k=|z_4{z}_1{z}_2|$