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

Question

For all complex numbers z1, z2 satisfying |z1|=12 and |z2-3-4 i|=5, the minimum value of |z1-z2| is (A) 0 (B) 2 (C) 7 (D) 17

Tardigrade Admin · Accepted Answer

By using property |z1-z2| ≥|z1|-|z2| |z1-z2|=|z1-(z2-3-4 i)-(3+4 i)| ≥|z1|-|z2-3-4 i|-|3+4 i| =12-5-5=2 |z1-z2| ≥ 2 Clearly from the figure: minimum value of |z1-z2|= AB = OB - OA =12-10=2 <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/22a14025c64b1487a8d5106fbf5bbbfd-.png" />

Q. For all complex numbers z1​,z2​ satisfying ∣z1​∣=12 and ∣z2​−3−4i∣=5, the minimum value of ∣z1​−z2​∣ is

0

2

7

17

By using property ∣z1​−z2​∣≥∣z1​∣−∣z2​∣ ∣z1​−z2​∣=∣z1​−(z2​−3−4i)−(3+4i)∣≥∣z1​∣−∣z2​−3−4i∣−∣3+4i∣ =12−5−5=2 ∣z1​−z2​∣≥2 Clearly from the figure : minimum value of ∣z1​−z2​∣=AB=OB−OA=12−10=2

Q. For all complex numbers $z_{1}, z_{2}$ satisfying $∣ z_{1} ∣ = 12$ and $∣ z_{2} - 3 - 4 i ∣ = 5$ , the minimum value of $∣ z_{1} - z_{2} ∣$ is