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
Step-by-Step Solution
Key Concept: Circle 1: centred at $(6,0)$, radius 5. Circle 2: centred at $(-2,6)$, radius 5. Let $z=x+iy$. From both circles: $x^2+y^2=12x-11$ and $x^2+y^2=-4x+12y-15$. Subtracting: $16x-12y+4=0\Rightarrow x=(3y-1)/4$.
$z=2+3i$. $z^3+3z^2-15z+141=50$.
Correct Answer: 1