Question

If $a,b,c$ are distincts & $w(\ne 1)$ is a cube of unity then minimum value of $x=|a+bw+c{w}^2|+|a+b{w}^2+cw|$

• A. $2\sqrt{3}$
• B. $3$
• C. $4\sqrt{2}$
• D. $2$

Answer 1

Answer: (A)

$2\sqrt{3}$

Let $z_1=b+bw+c{w}^2$
$|z_1{|}^2=|a+bw+c{w}^2{|}^2$
$\to z_1{z}_1=a^2+b^2+c^2-ab-bc-ca$
$\therefore |z_1|=\sqrt{a^2+b^2+c^2-ab-bc-ca}$
$\to \sqrt{\frac{1}{2}(a+b{)}^2+(b-c{)}^2+(c-a{)}^2}$
similarly $|z_2|=\sqrt{\frac{1}{2}\left\{\right(a-b{)}^2+(b-c{)}^2+(c-a{)}^2\}}$
$|z_1|+|z_2|=\sqrt{2}\sqrt{(a-b{)}^2+(b-c{)}^2+(c-a{)}^2}$
a, b, c are distinct integers
$\therefore$ min value$-1,0,1$
$|z_1|+|z_2|=\sqrt{1+1(2{)}^2}$
$|z_1|+|z_2|=\sqrt{2}\times \sqrt{6}=2\sqrt{3}$