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$If a + b + c = 0 then find the value of \frac{a^{2}}{bc} + \frac{b^{2}}{ac} + \frac{c^{2}}{ab}$

Subtopics: Algebraic Conditions

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Detailed Solution

\frac{a^{2}}{bc} + \frac{b^{2}}{ac} + \frac{c^{2}}{ab} = \frac{a^{2}}{bc} \frac{a}{a} + \frac{b^{2}}{ac} \frac{b}{b} + \frac{c^{2}}{ab} \frac{c}{c}

= \frac{a^{3}}{abc} + \frac{b^{3}}{abc} + \frac{c^{3}}{abc}

= \frac{a^{3} + b^{3} + c^{3}}{abc} - - - - - - - - - - - (1)

But we know that

{(a + b + c)}^{3} = a^{3} + b^{3} + c^{3} + 3 (a + b) (b + c) (a + c)

{(a + b + c)}^{3} - 3 (a + b) (b + c) (a + c) = a^{3} + b^{3} + c^{3} - - - - - - - - -

(1)
Given that

a + b + c = 0 then a + b = - c

Given that

a + b + c = 0 then a + c = - b

Given that

a + b + c = 0 then b + c = - a

Substitute above values in eqn (2)

0^{3} - 3 (- c) (- a) (- b) = a^{3} + b^{3} + c^{3}

0 + 3 abc = a^{3} + b^{3} + c^{3}

3 abc = a^{3} + b^{3} + c^{3}

Substitute above value in eqn(1)

\frac{a^{2}}{bc} + \frac{b^{2}}{ac} + \frac{c^{2}}{ab} = \frac{a^{3} + b^{3} + c^{3}}{abc} = \frac{3 abc}{abc} = 3

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