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JEE Main 2023 · 25 Jan · Shift 2

JEE Main 2023 Mathematics question on Complex Numbers and Quadratic Equations

Let zz be a complex number such that ∣z−2iz+i∣=2,z≠−i\left|\dfrac{z-2i}{z+i}\right| = 2, z \neq -i. Then zz lies on the circle of radius 2 and centre
  1. A.
    (0,2)(0, 2)
  2. B.
    (0,0)(0, 0)
  3. C.
    (2,0)(2, 0)
  4. D.
    (0,−2)(0, -2)
Show answer and solution

Correct option: D

Solution

Let z = x + iy. The condition |z - 2i| = 2|z + i| gives:

x² + (y-2)² = 4[x² + (y+1)²]

x² + y² - 4y + 4 = 4x² + 4y² + 8y + 4

3x² + 3y² + 12y = 0

x² + y² + 4y = 0

x² + (y+2)² = 4

This is a circle of radius 2 centred at (0, -2).

[JEE Main 2023 · 25 Jan · Shift 2]

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