For all complex numbers satisfying and , the minimum value of is
- A.0
- B.2
- C.7
- D.17
Show answer and solution
Correct option: B
Solution
|z₂ - (3+4i)| = 5 means z₂ lies on a circle centered at (3,4) with radius 5. Since |(3+4i)| = 5, this circle passes through the origin.
|z₁| = 12 means z₁ lies on a circle centered at origin with radius 12.
The minimum |z₁ - z₂| = |distance between centers - sum or difference of radii|. The center of z₂'s circle is at distance 5 from origin, and its radius is 5, so the closest point of z₂'s circle to the origin is 5 - 5 = 0 (it touches the origin). The farthest point is 5 + 5 = 10 from the origin.
The closest point on z₁'s circle to the origin side is at distance 12. The closest point on z₂'s circle (measured from origin along the same direction) reaches at most 10 from origin. So minimum |z₁ - z₂| = 12 - 10 = 2.
|z₁| = 12 means z₁ lies on a circle centered at origin with radius 12.
The minimum |z₁ - z₂| = |distance between centers - sum or difference of radii|. The center of z₂'s circle is at distance 5 from origin, and its radius is 5, so the closest point of z₂'s circle to the origin is 5 - 5 = 0 (it touches the origin). The farthest point is 5 + 5 = 10 from the origin.
The closest point on z₁'s circle to the origin side is at distance 12. The closest point on z₂'s circle (measured from origin along the same direction) reaches at most 10 from origin. So minimum |z₁ - z₂| = 12 - 10 = 2.
[AIEEE 2002]