Complex Numbers
Geometry of Complex Numbers
jee_main_2026_jan_21_shift_1
Grade None
Question:
If z is a complex number satisfying |z - 6| = 5 and |z + 2 - 6i| = 5, then the value of z³ + 3z² - 15z + 141 is equal to:
Step-by-Step Solution
Key Concept: Find z from intersection of two circles in complex plane.
Step 1: |z - 6| = 5 => circle center (6,0), radius 5. |z + 2 - 6i| = 5 => circle center (-2,6), radius 5. Step 2: Distance between centers = √(64+36) = 10. Intersection points: solving gives z = 2+3i. Step 3: z=2+3i. z² = -5+12i. z³ = -46+9i. Step 4: z³ + 3z² - 15z + 141 = (-46+9i) + 3(-5+12i) - 15(2+3i) + 141 = 50 + 0i = 50.
Correct Answer: D
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