Let z be a complex number such that |z-6|=5 and |z+2-6 i|=5 . Then the value of z³+3 z²-15 z+141 is equal to

Step-by-step solution

From |z - 6| = 5 , point z lies on circle centered at (6, 0) with radius 5. From |z + 2 - 6i| = 5 , point lies on circle centered at (-2, 6) with radius 5. Let z = x + iy . First condition gives: x^2 + y^2 = 12x - 11 . Second gives: x^2 + y^2 = -4x + 12y - 15 . Setting equal: 16x - 12y + 4 = 0 , so x = 3y - 1 4 . Substituting into (x-6)^2 + y^2 = 25 : (3y - 25)^2 + 16y^2 = 400 , which simplifies to 25y^2 - 150y + 225 = 0 , giving y = 3 and x = 2 . Thus z = 2 + 3i , so z^2 = -5 + 12i and z^3 = -46 + 9i . z^3 + 3z^2