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

Question

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

Accepted Answer

Center of first circle $C_{1}(6,0),r_{1} = 5$

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

$$\because C_{1}C_{2} = r_{1} + r_{2} $$∴ common point Z is mid point of $C_{1}$ & $C_{2}$

$${\therefore z = 2 + 3i }{\therefore z^{2} = 4z - 13 }{\therefore z^{3} = 3z - 52 }{\therefore z^{3} + 3z^{2} - 15z + 141 = 50}$$

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