Let $z_{1}$ and $z_{2}$ be two complex numbers satisfying $\left|z_{1}\right|=9$ and $\left|z_{2}-3-4 i\right|=4$. Then the minimum value of $\left|z_{1}-z_{2}\right|$ is :

JEE Main · 2019 · 12 Jan 2019 · Shift 2 · Complex Numbers

Let

z_{1}

and

z_{2}

be two complex numbers satisfying

∣ z_{1} ∣ = 9

and

∣ z_{2} - 3 - 4 i ∣ = 4

. Then the minimum value of

∣ z_{1} - z_{2} ∣

is :

Answer: (A)

z_{1}

lies on the circle of radius

9

about the origin, and

z_{2}

on the circle of radius

4

about

3 + 4 i

, whose centre is distance

∣3 + 4 i ∣ = 5

from the origin. Because

5 = 9 - 4

, the smaller circle sits exactly inside and touches the larger one internally, so the closest the two loci come is

9 - 4 - 5 = 0

JEE Main · 2019 · 12 Jan 2019 · Shift 2 · Single correct option