If a + b + c = 0, then write the value of a^2 bc + b^2 ca + c^2 a… - Vidyadip

Question

If $a + b + c = 0$, then write the value of $\frac{\text{a}^2}{\text{bc}}+\frac{\text{b}^2}{\text{ca}}+\frac{\text{c}^2}{\text{ab}}.$

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Answer

We have to find the value of $\frac{\text{a}^2}{\text{bc}}+\frac{\text{b}^2}{\text{ca}}+\frac{\text{c}^2}{\text{ab}}$
Given $a + b + c = 0$
Using identity $a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$
Put $a + b + c = 0 a^3 + b^3 + c^3 - 3abc = (0)(a^2 + b^2 + c^2 - ab - bc - ca) a^3 + b^3 + c^3 - 3$
$abc$ $= 0 a^3 + b^3 + c^3 = 3abc$
$\frac{\text{a}^3}{\text{abc}}+\frac{\text{b}^3}{\text{abc}}+\frac{\text{c}^3}{\text{abc}}=3$
$\frac{\text{a}\times\text{a}\times\text{a}}{\text{abc}}+\frac{\text{b}\times\text{b}\times\text{b}}{\text{abc}}+\frac{\text{c}\times\text{c}\times\text{c}}{\text{abc}}=3$
$\frac{\text{a}^2}{\text{bc}}+\frac{\text{b}^2}{\text{ca}}+\frac{\text{c}^2}{\text{ab}}=3$
Hence the value of $\frac{\text{a}^2}{\text{bc}}+\frac{\text{b}^2}{\text{ca}}+\frac{\text{c}^2}{\text{ab}}$ is $3$.

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