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Find \(ab + bc + ca\) given that:

\(a^{2} + ab + b^{2} = 2\)

\(b^{2} + bc + c^{2} = 1\)

\(c^{2} + ca + a^{2} = 3\)

Giveb that a, b, and c are positive numbers.

Note by Christian Daang
1 year, 3 months ago

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\(a^2+ab+b^2=2....Eq.1\)

\(b^2+bc+c^2=1....Eq.2\)

\(c^2+ac+a^2=3....Eq.3\)

Eq.1-Eq.2

\( (a+b+c)(a-c)=1...Eq.4\)

Eq.3-Eq.2

\( (a+b+c)(a-b)=1...Eq.5\)

Eq.3-Eq.1

\( (a+b+c)(c-b)=1...Eq.6\)

Comparing Eq.4,5,6, we get \(a=b=c\)

Substituting these values we get an inconsistent system and hence we conclude that there is no real solution. Aneesh Kundu · 1 year, 3 months ago

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Is the question correct? There don't exist positive \(a,b,c \) for which these equations are true. Siddhartha Srivastava · 1 year, 3 months ago

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@Siddhartha Srivastava Yes sir, there is. :) Christian Daang · 1 year, 3 months ago

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