If the roots of the equation `(a^(2) + b^(2)) x^(2) - 2b ( a+c) x+ ( b^(2) + c^(2)) = 0 ` are equal , then which one of the following is correct ?

Options

A. ` ab = a+c`
B. ` b^(2) = ac`
C. ` b+c=2a`
D. `b=ac`

Explanation

To solve the problem, we need to determine the condition under which the roots of the quadratic equation

\[ (a^2 + b^2)x^2 - 2b(a + c)x + (b^2 + c^2) = 0 \]

are equal. For a quadratic equation of the form \(Ax^2 + Bx + C = 0\), the roots are equal if the discriminant \(D\) is zero. The discriminant is given by:

\[ D = B^2 - 4AC \]

### Step 1: Identify coefficients
From the given equation, we identify the coefficients:
- \(A = a^2 + b^2\)
- \(B = -2b(a + c)\)
- \(C = b^2 + c^2\)

### Step 2: Write the discriminant
Now, we can write the discriminant \(D\):

\[ D = B^2 - 4AC \]

Substituting the values of \(A\), \(B\), and \(C\):

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

### Step 3: Simplify the discriminant
Calculating \(B^2\):

\[ B^2 = 4b^2(a + c)^2 \]

Now substituting this back into the discriminant:

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

### Step 4: Factor out common terms
We can factor out \(4\) from the discriminant:

\[ D = 4\left[b^2(a + c)^2 - (a^2 + b^2)(b^2 + c^2)\right] \]

### Step 5: Set the discriminant to zero
For the roots to be equal, we set \(D = 0\):

\[ b^2(a + c)^2 - (a^2 + b^2)(b^2 + c^2) = 0 \]

### Step 6: Expand and simplify
Expanding both sides:

\[ b^2(a^2 + 2ac + c^2) = a^2b^2 + b^4 + b^2c^2 \]

Rearranging gives:

\[ b^2a^2 + 2abc^2 + b^2c^2 = a^2b^2 + b^4 + b^2c^2 \]

### Step 7: Cancel and simplify
Canceling \(b^2c^2\) and \(a^2b^2\) from both sides:

\[ 2abc^2 = b^4 \]

### Step 8: Rearranging the equation
Rearranging gives:

\[ b^2 = ac \]

### Conclusion
Thus, the condition for the roots of the given quadratic equation to be equal is:

\[ b^2 = ac \]