If `P = (4xy)/(x+y)`, find the valuee of `(P + 2x) / (P-2x) + (P + 2y)/(P-2y)`

Options

A. 4
B. 1
C. 2
D. 6

Answer

To solve the problem, we need to find the value of

\[ \frac{P + 2x}{P - 2x} + \frac{P + 2y}{P - 2y} \]

given that

\[ P = \frac{4xy}{x + y}. \]

### Step 1: Substitute the value of P

First, we substitute the value of \( P \) into the expression:

\[ \frac{\frac{4xy}{x+y} + 2x}{\frac{4xy}{x+y} - 2x} + \frac{\frac{4xy}{x+y} + 2y}{\frac{4xy}{x+y} - 2y}. \]

### Step 2: Simplify the first term

For the first term:

\[ \frac{\frac{4xy}{x+y} + 2x}{\frac{4xy}{x+y} - 2x} \]

We can combine the fractions in the numerator and denominator:

Numerator:
\[ \frac{4xy + 2x(x+y)}{x+y} = \frac{4xy + 2x^2 + 2xy}{x+y} = \frac{6xy + 2x^2}{x+y} \]

Denominator:
\[ \frac{4xy - 2x(x+y)}{x+y} = \frac{4xy - 2x^2 - 2xy}{x+y} = \frac{2xy - 2x^2}{x+y} \]

Thus, the first term simplifies to:

\[ \frac{6xy + 2x^2}{2xy - 2x^2} = \frac{2(3xy + x^2)}{2(y - x^2)} = \frac{3xy + x^2}{y - x^2} \]

### Step 3: Simplify the second term

Now, for the second term:

\[ \frac{\frac{4xy}{x+y} + 2y}{\frac{4xy}{x+y} - 2y} \]

Using the same method:

Numerator:
\[ \frac{4xy + 2y(x+y)}{x+y} = \frac{4xy + 2xy + 2y^2}{x+y} = \frac{6xy + 2y^2}{x+y} \]

Denominator:
\[ \frac{4xy - 2y(x+y)}{x+y} = \frac{4xy - 2xy - 2y^2}{x+y} = \frac{2xy - 2y^2}{x+y} \]

Thus, the second term simplifies to:

\[ \frac{6xy + 2y^2}{2xy - 2y^2} = \frac{2(3xy + y^2)}{2(xy - y^2)} = \frac{3xy + y^2}{xy - y^2} \]

### Step 4: Combine both terms

Now we have:

\[ \frac{3xy + x^2}{y - x^2} + \frac{3xy + y^2}{xy - y^2} \]

To combine these fractions, we need a common denominator, which is \((y - x^2)(xy - y^2)\).

### Step 5: Final expression

After finding the common denominator and combining, we will simplify the expression to get the final result.

### Final Result

The final value of

\[ \frac{P + 2x}{P - 2x} + \frac{P + 2y}{P - 2y} \]

is

\[ \frac{(3xy + x^2)(xy - y^2) + (3xy + y^2)(y - x^2)}{(y - x^2)(xy - y^2)}. \]