benchmarks.wiki

benchmarks.wiki / Public workspace

MATH / Let z₁=18+83i, z₂=18+39i, and z₃=78+99i, where i²=−1. Let z be the unique complex number with the properties that (z₃−z₁)/(z₂−z₁)⋅(z−z₂)/(z−z₃) is a real number and the imaginary p…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Let z1=18+83iz_1 = 18 + 83i, z2=18+39i,z_2 = 18 + 39i, and z3=78+99i,z_3 = 78 + 99i, where i2=−1i^2 = -1. Let zz be the unique complex number with the properties that z3−z1z2−z1⋅z−z2z−z3\frac{z_3 - z_1}{z_2 - z_1} \cdot \frac{z - z_2}{z - z_3} is a real number and the imaginary part of zz is the greatest possible. Find the real part of zz.
Plain-text mathematical notation (without MathML)
Let z₁=18+83i, z₂=18+39i, and z₃=78+99i, where i²=−1. Let z be the unique complex number with the properties that (z₃−z₁)/(z₂−z₁)⋅(z−z₂)/(z−z₃) is a real number and the imaginary part of z is the greatest possible. Find the real part of z.
Original LaTeX notation
Let $z_1 = 18 + 83i$, $z_2 = 18 + 39i,$ and $z_3 = 78 + 99i,$ where $i^2 = -1$. Let $z$ be the unique complex number with the properties that $\frac{z_3 - z_1}{z_2 - z_1} \cdot \frac{z - z_2}{z - z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.

level

Level 5

type

Intermediate Algebra

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
9f0842d5-e788-5991-9a41-b541de8b73b0
task_key
intermediate~5falgebra--test--9f0842d5-e788-5991-9a41-b541de8b73b0
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import