If `a+b+c=0`, then the value of `(a^2 + b^2 +c^2)/(bc+ca+ab)` is….

Answer

To solve the problem, we start with the given condition:

**Given:**
\( a + b + c = 0 \)

We need to find the value of:

\[ \frac{a^2 + b^2 + c^2}{bc + ca + ab} \]

**Step 1: Use the identity for the square of a sum.**
We know that:

\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]

Since \( a + b + c = 0 \), we can substitute this into the identity:

\[ 0^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]

This simplifies to:

\[ 0 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]

**Step 2: Rearranging the equation.**
Rearranging the equation gives us:

\[ a^2 + b^2 + c^2 = -2(ab + bc + ca) \]

**Step 3: Substitute into the original expression.**
Now we substitute \( a^2 + b^2 + c^2 \) into our expression:

\[ \frac{a^2 + b^2 + c^2}{bc + ca + ab} = \frac{-2(ab + bc + ca)}{bc + ca + ab} \]

**Step 4: Simplify the expression.**
We can simplify this expression:

\[ = -2 \cdot \frac{(ab + bc + ca)}{(ab + bc + ca)} \]

As long as \( ab + bc + ca \neq 0 \), this simplifies to:

\[ = -2 \]

Thus, the final value is:

\[ \boxed{-2} \]

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