Let z be a complex number such that | z - 6 | = 5 a n d | z + 2 - 6 i | = 5 . Then the value of z 3 + 3 z 2 - 15 z + 141 is equal to [2026]

Question

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 [2026]

Smart Achievers · Accepted Answer

(2)

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

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

$∵$ $C_{1} C_{2} = r_{1} + r_{2}$

$∴$ $common point Z is mid point of C_{1} & C_{2}$

$∴$ $z = 2 + 3 i$

$∴$ $z^{2} = 4 z - 13$

$∴$ $z^{3} = 3 z - 52$

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

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