Let z be a complex number such that $|\mathrm{z}-6|=5$ and $|z+2-6 i|=5$. Then the value of $z^{3}+3 z^{2}-15 z+141$ is equal to

JEE Main · 2026 · 28 Jan 2026 · Shift 1 · Complex Numbers

Let z be a complex number such that

∣ z - 6∣ = 5

and

∣ z + 2 - 6 i ∣ = 5

.
Then the value of

z^{3} + 3 z^{2} - 15 z + 141

is equal to

Answer: (C)

The conditions describe circles of radius

5

centred at

(6, 0)

and

(- 2, 6)

. The distance between the centres is

8^{2} + 6^{2} = 10 = 5 + 5

, so the circles touch externally at the single point dividing the join of the centres equally, namely

z = \frac{6 + ( - 2 )}{2} + i \frac{0 + 6}{2} = 2 + 3 i

. Then

z^{2} = - 5 + 12 i

and

z^{3} = - 46 + 9 i

, so

z^{3} + 3 z^{2} - 15 z + 141 = (- 46 + 9 i) + (- 15 + 36 i) - (30 + 45 i) + 141 = 50

JEE Main · 2026 · 28 Jan 2026 · Shift 1 · Single correct option