Let z _ 1 and z _ 2 be two complex numbers satisfying | z _ 1 | = 9...

We have,

${C}_{1}: \left|{z}_{1}\right| = 9$

${C}_{2}: \left|{z}_{2} - 3 - 4i\right| = 4$

Centres of the circles are ${c}_{1} ≡$\left(0, 0\right)$and${c}_{2} ≡ \left(3, 4\right)$.</p><p>Radius of the circles are${r}_{1} = 9$and${r}_{2} = 4$.</p><p>$⇒ {c}_{1}{c}_{2} = \sqrt{{3}^{2} + {4}^{2}} = 5$and${r}_{1} - {r}_{2} = 5$</p><p>Circles touch each other internally.<br /><br />$⇒ {\left|{z}_{1} - {z}_{2}\right|}_\min = 0 at the point of contact.