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MATH / There exist two complex numbers c, say c₁ and c₂, so that −5+3i, 8−i, and c form the vertices of an equilateral triangle. Find the product c₁c₂. [asy] unitsize(0.5 cm); pair A, B; …

Problem

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

problem

There exist two complex numbers cc, say c1c_1 and c2c_2, so that −5+3i-5 + 3i, 8−i8 - i, and cc form the vertices of an equilateral triangle. Find the product c1c2c_1 c_2. [asy] unitsize(0.5 cm); pair A, B; pair[] C; A = (2,2); B = (5,1); C[1] = rotate(60,A)*(B); C[2] = rotate(60,B)*(A); draw(A--C[1]--B--C[2]--cycle); draw(A--B); dot("−5+3i-5 + 3i", A, W); dot("8−i8 - i", B, E); dot("c1c_1", C[1], N); dot("c2c_2", C[2], S); [/asy]
Plain-text mathematical notation (without MathML)
There exist two complex numbers c, say c₁ and c₂, so that −5+3i, 8−i, and c form the vertices of an equilateral triangle.  Find the product c₁c₂.

[asy]
unitsize(0.5 cm);

pair A, B;
pair[] C;

A = (2,2);
B = (5,1);
C[1] = rotate(60,A)*(B);
C[2] = rotate(60,B)*(A);

draw(A--C[1]--B--C[2]--cycle);
draw(A--B);

dot("−5+3i", A, W);
dot("8−i", B, E);
dot("c₁", C[1], N);
dot("c₂", C[2], S);
[/asy]
Original LaTeX notation
There exist two complex numbers $c$, say $c_1$ and $c_2$, so that $-5 + 3i$, $8 - i$, and $c$ form the vertices of an equilateral triangle.  Find the product $c_1 c_2$.

[asy]
unitsize(0.5 cm);

pair A, B;
pair[] C;

A = (2,2);
B = (5,1);
C[1] = rotate(60,A)*(B);
C[2] = rotate(60,B)*(A);

draw(A--C[1]--B--C[2]--cycle);
draw(A--B);

dot("$-5 + 3i$", A, W);
dot("$8 - i$", B, E);
dot("$c_1$", C[1], N);
dot("$c_2$", C[2], S);
[/asy]

level

Level 5

type

Precalculus

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Source and history

task_id
4cfc045a-6bc6-5e7f-8754-56181da5421a
task_key
precalculus--test--4cfc045a-6bc6-5e7f-8754-56181da5421a
task_revision_id
1
config / split
precalculus / test

Official source

initial import