If `a`, `b`, `c` are all non-zero and `a+b+c=0`, then `(a^(2))/(bc)+(b^(2))/(ca)+(c^(2))/(ab)= ?`.

Options

A. 1
B. 2
C. 3
D. 4

Explanation

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

**Given:**
\[ a + b + c = 0 \]

We need to find the value of:
\[ \frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab} \]

### Step 1: Use the identity for cubes
We can use the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]

Since we know \( a + b + c = 0 \), we can simplify the identity:
\[ a^3 + b^3 + c^3 - 3abc = 0 \]
This implies:
\[ a^3 + b^3 + c^3 = 3abc \]

### Step 2: Divide by \( abc \)
Now, we divide both sides of the equation by \( abc \):
\[ \frac{a^3}{abc} + \frac{b^3}{abc} + \frac{c^3}{abc} = 3 \]

### Step 3: Simplify the fractions
This simplifies to:
\[ \frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab} = 3 \]

### Conclusion
Thus, we have found that:
\[ \frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab} = 3 \]

### Final Answer
The value is:
\[ \boxed{3} \]

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