For $z \in \mathbb{C}$ if the minimum value of $(|z-3 \sqrt{2}|+|z-p \sqrt{2} i|)$ is $5 \sqrt{2}$, then a value of $p$ is

JEE Main · 2022 · 25 Jul 2022 · Shift 2 · Complex Numbers

For

z \in C

if the minimum value of

(∣ z - 32 ∣ + ∣ z - p 2 i ∣)

52

, then a value of

p

Answer: (C)

The minimum value of

∣ z - z_{1} ∣ + ∣ z - z_{2} ∣

occurs when

z

lies on the line segment joining

z_{1}

and

z_{2}

, and equals

∣ z_{1} - z_{2} ∣

. Here

z_{1} = 32

and

z_{2} = p 2 i

. The minimum value is

∣3 2 - p 2 i ∣ = (32)^{2} + (p 2)^{2} = 18 + 2 p^{2} = 2 9 + p^{2}

. Setting this equal to

52

2 9 + p^{2} = 52

, so

9 + p^{2} = 5

, giving

9 + p^{2} = 25

, hence

p^{2} = 16

and

p = \pm 4

. The answer is

p = 4

JEE Main · 2022 · 25 Jul 2022 · Shift 2 · Single correct option