Question

If $|z_1-1|<1,|z_2-2|<2,|z_3-3|<3,then|z_1+z_2+z_3|$

• A. is less than $6$
• B. is more than $3$
• C. is less than $12$
• D. lies between $6$ and $12$

Answer 1

The correct option is C is less than $12$

$|z_1-1|<1,$ $|z_2-2|<2,$ $|z_3-3|<3,$

$\Rightarrow |z_1+z_2+z_3|=?$

$|z_1-1|<1$

$\Rightarrow z_1=x_1+i{y}_2$ $z_2=x_2+i{y}_2$

$\Rightarrow \sqrt{{(x_1-1)}^2+{\left(iy\right)}^2}=1$ $\sqrt{{(x_2-2)}^2+{\left(iy\right)}^2}=2$

$\Rightarrow x_1^2-2x_1+1-y_1^2=1$ ${(x_2-2)}^2-y_2^2=4$

$x_1^2{-y}_1^2=2x_1$ $x_2^2{-4x_2+4-y}_2^2=4$

$x_1^2-2x_1-y_1^2=0$ ${(x_2-2)}^2-y_2^2=0$

${(x_1-1)}^2-y_1^2=0$ $x_2=2$

$\Rightarrow x_1=1$

Similarly, $z_3=3$

$\Rightarrow |z_1+z_2+z_3|=|1+2+3|=\left|6\right|$

$\therefore$ The value is less then $6.$

Hence, the answer is less than $12$.