HMMT 十一月 2018 · 冲刺赛 · 第 20 题

HMMT November 2018 — Guts Round — Problem 20

[ 11 ] Let z be a complex number. In the complex plane, the distance from z to 1 is 2, and the distance 2 from z to 1 is 6. What is the real part of z ?

解析

[ 11 ] Let z be a complex number. In the complex plane, the distance from z to 1 is 2, and the distance 2 from z to 1 is 6. What is the real part of z ? Proposed by: James Lin 5 Answer: 4 2 | z 1 | 2 Note that we must have | z 1 | = 2 and | z 1 | = 6, so | z + 1 | = = 3. Thus, the distance from z | z 1 | to 1 in the complex plane is 2 and the distance from z to 1 in the complex plane is 3. Thus, z, 1 , 1 form a triangle with side lengths 2 , 3 , 3. The area of a triangle with sides 2 , 2 , 3 can be computed to p p p 3 7 3 7 2 3 7 be by standard techniques, so the length of the altitude from z to the real axis is · = . 4 4 2 4 r ⇣ ⌘ p 2 3 7 1 2 The distance between 1 and the foot from z to the real axis is 2 = by the Pythagorean 4 4 Theorem. It is clear that z has positive imaginary part as the distance from z to 1 is greater than 1 5 the distance from z to 1, so the distance from 0 to the foot from z to the real axis is 1 + = . This 4 4 is exactly the real part of z that we are trying to compute.