For `a`, `b`,`c` non-zero, real distinct, the equation, `(a^(2)+b^(2))x^(2)-2b(a+c)x+b^(2)+c^(2)=0` has non-zero real roots. One of these roots is also the root of the equation :

Options

A. `(b^(2)-c^(2))x^(2)+2a(b-c)x-a^(2)=0`
B. `(b^(2)+c^(2))x^(2)-2a(b+c)x+a^(2)=0`
C. `a^(2)x^(2)+a(c-b)x-bc=0`
D. `a^(2)x^(2)-a(b-c)x+bc=0`

Explanation

To solve the given equation and find the correct option, we will follow these steps:

### Step 1: Identify the given quadratic equation
The given equation is:
\[ (a^2 + b^2)x^2 - 2b(a + c)x + (b^2 + c^2) = 0 \]

### Step 2: Calculate the discriminant
For the roots to be real and distinct, the discriminant \(D\) of the quadratic equation must be greater than zero. The discriminant \(D\) is given by:
\[ D = B^2 - 4AC \]
where \(A = a^2 + b^2\), \(B = -2b(a + c)\), and \(C = b^2 + c^2\).

Calculating \(D\):
\[ D = (-2b(a + c))^2 - 4(a^2 + b^2)(b^2 + c^2) \]
\[ D = 4b^2(a + c)^2 - 4(a^2 + b^2)(b^2 + c^2) \]

### Step 3: Simplify the discriminant
Factoring out 4 from the discriminant:
\[ D = 4\left[b^2(a + c)^2 - (a^2 + b^2)(b^2 + c^2)\right] \]

### Step 4: Set the discriminant greater than zero
For the roots to be real and distinct:
\[ b^2(a + c)^2 - (a^2 + b^2)(b^2 + c^2) > 0 \]

### Step 5: Solve for the roots
The roots of the quadratic equation can be found using the quadratic formula:
\[ x = \frac{-B \pm \sqrt{D}}{2A} \]
Substituting \(B\) and \(D\):
\[ x = \frac{2b(a + c) \pm \sqrt{4\left[b^2(a + c)^2 - (a^2 + b^2)(b^2 + c^2)\right]}}{2(a^2 + b^2)} \]
\[ x = \frac{b(a + c) \pm \sqrt{b^2(a + c)^2 - (a^2 + b^2)(b^2 + c^2)}}{a^2 + b^2} \]

### Step 6: Identify one of the roots
Let’s denote one of the roots as \(x_1\):
\[ x_1 = \frac{b(a + c) + \sqrt{b^2(a + c)^2 - (a^2 + b^2)(b^2 + c^2)}}{a^2 + b^2} \]

### Step 7: Check which option has \(x_1\) as a root
Now we need to check which of the given options has \(x_1\) as a root. We will substitute \(x_1\) into each option and see if it satisfies the equation.

### Step 8: Evaluate the options
1. **Option 1**: \(b^2 - c^2)x^2 + 2b - c)x - a^2 = 0\)
2. **Option 2**: \(b^2 + c^2)x^2 + 2b + c)x + a^2 = 0\)
3. **Option 3**: \(a^2x^2 + ab - c)x - bc = 0\)
4. **Option 4**: \(a^2x^2 - ab - c)x + bc = 0\)

After checking, we find that **Option 3** matches with our derived root.

### Final Answer:
The correct option is:
\[ \text{Option 3: } a^2x^2 + ab - c)x - bc = 0 \]

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