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  4. For all complex numbers z1, z2 satisfying |z1|=12 and |z2-3-4 i|=5, the minimum value of |z1-z2| is

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Q. For all complex numbers $z_1, z_2$ satisfying $\left|z_1\right|=12$ and $\left|z_2-3-4 i\right|=5$, the minimum value of $\left|z_1-z_2\right|$ is

Complex Numbers and Quadratic Equations

Solution:

By using property $\left|z_1-z_2\right| \geq\left|z_1\right|-\left|z_2\right|$
$ \left|z_1-z_2\right|=\left|z_1-\left(z_2-3-4 i\right)-(3+4 i)\right| \geq\left|z_1\right|-\left|z_2-3-4 i\right|-|3+4 i|$
$=12-5-5=2$
$ \left|z_1-z_2\right| \geq 2$
Clearly from the figure : minimum value of
$\left|z_1-z_2\right|= AB = OB - OA =12-10=2$
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