If `sqrt(x+y) +sqrt(y-x)=5,` then ` (d^(2)y)/(dx ^(2))=`

Options

A. `2/5`
B. `4/25`
C. `2/25`
D. `1/25`

Answer

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

\[ \sqrt{x+y} + \sqrt{y-x} = 5 \]

### Step 1: Square both sides
First, we square both sides of the equation to eliminate the square roots:

\[ (\sqrt{x+y} + \sqrt{y-x})^2 = 5^2 \]

Expanding the left side, we have:

\[ x+y + y-x + 2\sqrt{(y-x)(x+y)} = 25 \]

### Step 2: Simplify the equation
Now, simplify the left side:

\[ 2y + 2\sqrt{(y-x)(x+y)} = 25 \]

Next, isolate the square root term:

\[ 2\sqrt{(y-x)(x+y)} = 25 - 2y \]

### Step 3: Square again
Now, we square both sides again to eliminate the square root:

\[ 4(y-x)(x+y) = (25 - 2y)^2 \]

### Step 4: Expand both sides
Expanding both sides gives:

\[ 4(y^2 - x^2) = 625 - 100y + 4y^2 \]

### Step 5: Rearranging the equation
Rearranging the equation leads to:

\[ 4y^2 - 4x^2 - 625 + 100y - 4y^2 = 0 \]

This simplifies to:

\[ -4x^2 + 100y - 625 = 0 \]

### Step 6: Differentiate with respect to x
Now we differentiate this equation with respect to \(x\):

\[ \frac{d}{dx}(-4x^2) + \frac{d}{dx}(100y) - \frac{d}{dx}(625) = 0 \]

This results in:

\[ -8x + 100\frac{dy}{dx} = 0 \]

### Step 7: Solve for \(\frac{dy}{dx}\)
From this, we can solve for \(\frac{dy}{dx}\):

\[ 100\frac{dy}{dx} = 8x \]
\[ \frac{dy}{dx} = \frac{8x}{100} = \frac{2x}{25} \]

### Step 8: Differentiate again to find \(\frac{d^2y}{dx^2}\)
Next, we differentiate \(\frac{dy}{dx}\) again with respect to \(x\):

\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{2x}{25}\right) = \frac{2}{25} \]

### Final Answer
Thus, the value of \(\frac{d^2y}{dx^2}\) is:

\[ \frac{d^2y}{dx^2} = \frac{2}{25} \]

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