Question

Let w₁ and w₂ be two complex numbers satisfying |w₁| = 12 and |w₂ – 5 – 12i| = 25. Then, the minimum value of |w₁ – w₂| is

• A. 0
• B. 5
• C. 13
• D. 12

Answer 1

Answer: (A)

0

Answer 2

$|w_1| = 12$ implies $w_1$ is on a circle $C_1$ with center $c_1 = 0$ and radius $r_1 = 12$. $|w_2 - (5 + 12i)| = 25$ implies $w_2$ is on a circle $C_2$ with center $c_2 = 5 + 12i$ and radius $r_2 = 25$.

The distance between centers is $d = |c_1 - c_2| = |0 - (5 + 12i)| = |-(5 + 12i)| = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.

The absolute difference of the radii is $|r_1 - r_2| = |12 - 25| = |-13| = 13$.

Since $d = |r_1 - r_2|$, the circles touch internally. The minimum distance between points on internally tangent circles is 0.