Solve the equation :
`(a)/(x-b)+(b)/(x-a)=2" "(xneb,a)`

Options

A. `x = a-b and x= (a-b) / (2)`
B. `x = a-b and x= (a+b) / (2)`
C. `x = a+b and x= (a-b) / (2)`
D. `x = a+b and x= (a+b) / (2)`

Explanation

To solve the equation

\[ \frac{a}{x-b} + \frac{b}{x-a} = 2 \]

where \( x \neq a \) and \( x \neq b \), we can follow these steps:

### Step 1: Find a common denominator
The common denominator for the left side of the equation is \((x-b)(x-a)\). We can rewrite the equation as:

\[ \frac{a(x-a) + b(x-b)}{(x-b)(x-a)} = 2 \]

### Step 2: Multiply both sides by the common denominator
Multiplying both sides by \((x-b)(x-a)\) gives:

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

### Step 3: Expand both sides
Now, we will expand both sides:

Left side:
\[ a(x-a) + b(x-b) = ax - a^2 + bx - b^2 = (a+b)x - (a^2 + b^2) \]

Right side:
\[ 2(x-b)(x-a) = 2(x^2 - (a+b)x + ab) = 2x^2 - 2(a+b)x + 2ab \]

### Step 4: Set the equation to zero
Now we can set the equation to zero by moving all terms to one side:

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

This simplifies to:

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

Combining like terms gives:

\[ -2x^2 + (3(a+b))x + (2ab - a^2 - b^2) = 0 \]

### Step 5: Rearranging the equation
Rearranging gives us:

\[ 2x^2 - 3(a+b)x + (a^2 + b^2 - 2ab) = 0 \]

### Step 6: Use the quadratic formula
Now we can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 2 \), \( b = -3(a+b) \), and \( c = a^2 + b^2 - 2ab \):

\[ x = \frac{3(a+b) \pm \sqrt{(-3(a+b))^2 - 4 \cdot 2 \cdot (a^2 + b^2 - 2ab)}}{2 \cdot 2} \]

### Step 7: Simplifying the discriminant
Calculating the discriminant:

\[ (-3(a+b))^2 = 9(a+b)^2 \]
\[ 4 \cdot 2 \cdot (a^2 + b^2 - 2ab) = 8(a^2 + b^2 - 2ab) = 8((a-b)^2) \]

So the discriminant simplifies to:

\[ 9(a+b)^2 - 8(a-b)^2 \]

### Step 8: Final solution
Substituting back into the quadratic formula gives us the two possible solutions for \( x \):

1. \( x = \frac{3(a+b) + \sqrt{9(a+b)^2 - 8(a-b)^2}}{4} \)
2. \( x = \frac{3(a+b) - \sqrt{9(a+b)^2 - 8(a-b)^2}}{4} \)