Question

Let z satisfy $\left|z-3\right|=4$ ,then find max. and min. value of $\left|z-5-7i\right|.$

Answer 1

given that

$|z-3|=4$

then

$|z-5-7i|=|z-3-2-7i|$

$=|z-3-(2+7i\left)\right|$

$=||z-3|+|2+7i||$ $\{\because |z_1-z_2|=\left|z_1\right|+\left|z_2\right|\}$

for maximum value

$=4+\sqrt{4+49}$

$=4+\sqrt{53}$

for min. value

$=|4-\sqrt{4+49}|$

$=|4-\sqrt{53}|$