Question

$|Z_1|=2$ and $|Z_2-6-8i|=4$, then the minimum distance between $Z_1$ and $Z_2$ is:

• A. $10$
• B. $4$
• C. $2$
• D. $16$

Answer 1

The correct option is B $4$

Given $\left|z_1\right|=2⟶\left(1\right)$

$\Rightarrow |z_2-6-8i|=4⟶\left(2\right)$

replace $z_1\xi z_2$ by $z=x+iy$

$\left(1\right)\Rightarrow x^2+y^2=4⟶\left(3\right)$

$\left(2\right)\Rightarrow {(x-6)}^2+{(y-8)}^2=16⟶\left(4\right)$

Are two circles with $c_1(0,0),c_2(6,8)$ as centers and $2,$ $4$ as radius respectively.

Two are not intersecting circles as $r_1+r_2<c_1{c}_2$

$\Rightarrow$ Minimum distance is $c_1{c}_2-r_1-r_2=10-2-4=4$

Maximum distance is $c_1{c}_2+r_1+r_2=10+2+4=16$

Hence, the answer is $4$