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JEE Main 2019 · 12 Jan · Shift II

JEE Main 2019 Mathematics question on Complex Numbers and Quadratic Equations

Let z1z_1 and z2z_2 be two complex numbers satisfying ∣z1∣=9|z_1| = 9 and ∣z2−3−4i∣=4|z_2 - 3 - 4i| = 4. Then, the minimum value of ∣z1−z2∣|z_1 - z_2| is
  1. A.
    1
  2. B.
    2
  3. C.
    2\sqrt{2}
  4. D.
    0
Show answer and solution

Correct option: D

Solution

The center of z₂'s locus is 3+4i, which has modulus |3+4i| = 5, and z₂ lies on a circle of radius 4 centered there. So z₂ ranges from distance 5-4=1 to 5+4=9 from the origin. Since z₁ lies on the circle |z₁|=9, z₂ can also reach distance 9 from the origin (at the point 9·(3+4i)/5 = 27/5 + 36i/5... wait, max distance = 5+4=9). So both z₁ and z₂ can simultaneously equal the same point at distance 9 from origin on the ray toward 3+4i, giving |z₁-z₂|=0.

[JEE Main 2019 · 12 Jan · Shift II]

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