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

Mathematics · Complex Numbers

JEE Main 2026 — 28 January, Morning Shift — Question 5

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

Center of first circle $\mathrm{C}_{1}(6,0), \mathrm{r}_{1}=5$

Center of second circle $C_{2}(-2,6), r_{2}=5$

$\because \mathrm{C}_{1} \mathrm{C}_{2}=\mathrm{r}_{1}+\mathrm{r}_{2}$

∴ common point Z is mid point of $\mathrm{C}_{1}$ & $\mathrm{C}_{2}$

$\therefore \mathrm{z}=2+3 \mathrm{i}$

$\therefore \mathrm{z}^{2}=4 \mathrm{z}-13$

$\therefore z^{3}=3 z-52$

$\therefore z^{3}+3 z^{2}-15 z+141=50$

Answer key and solution verified before publishing.

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