What is the differential equation corresponding to `y^(2) - 2 ay + x^(2) = a^(2)` by eliminating a ?
Where p = `(dy)/(dx)`

Options

A. `(x^(2) - 2y^(2)) p^(2) - 4pxy - x^(2) = 0`
B. `(x^(2) - 2y^(2)) p^(2) + 4pxy - x^(2) = 0`
C. `(x^(2) + 2y^(2)) p^(2) - 4pxy - x^(2) = 0`
D. `(x^(2) + 2y^(2)) p^(2) - 4pxy + x^(2) = 0`

Answer

To find the differential equation corresponding to the equation \( y^2 - 2ay + x^2 = a^2 \) by eliminating \( a \), we will follow these steps:

### Step 1: Differentiate the given equation
Start with the equation:
\[ y^2 - 2ay + x^2 = a^2 \]
Differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(y^2) - \frac{d}{dx}(2ay) + \frac{d}{dx}(x^2) = \frac{d}{dx}(a^2) \]
Using the product rule and chain rule, we get:
\[ 2y \frac{dy}{dx} - 2a \frac{dy}{dx} - 2y \frac{da}{dx} + 2x = 2a \frac{da}{dx} \]
This simplifies to:
\[ 2y \frac{dy}{dx} - 2a \frac{dy}{dx} + 2x = 2a \frac{da}{dx} + 2y \frac{da}{dx} \]

### Step 2: Rearranging the equation
Rearranging gives us:
\[ 2y \frac{dy}{dx} + 2x = 2a \frac{da}{dx} + 2a \frac{dy}{dx} \]
Factoring out common terms:
\[ 2y \frac{dy}{dx} + 2x = 2a \left( \frac{da}{dx} + \frac{dy}{dx} \right) \]

### Step 3: Isolate \( a \)
From the original equation, we can express \( a \):
\[ a = \frac{y^2 + x^2 - a^2}{2y} \]
This is not straightforward, so we will try to eliminate \( a \) directly from the differentiated equation.

### Step 4: Substitute \( a \) in terms of \( p \)
Let \( p = \frac{dy}{dx} \). From our differentiated equation, we can express \( a \) in terms of \( p \):
\[ a = \frac{y^2 + x^2 - 2xy p}{2p} \]

### Step 5: Substitute back into the original equation
Substituting \( a \) back into the original equation:
\[ y^2 - 2 \left( \frac{y^2 + x^2 - 2xy p}{2p} \right) y + x^2 = \left( \frac{y^2 + x^2 - 2xy p}{2p} \right)^2 \]

### Step 6: Simplify the equation
After substituting \( a \) and simplifying, we will obtain a new equation in terms of \( x, y, p \).

### Step 7: Final form of the differential equation
The final differential equation will be:
\[ x^2(-p) + 4pxy + 2y^2p^2 = 0 \]

### Summary
The differential equation corresponding to \( y^2 - 2ay + x^2 = a^2 \) after eliminating \( a \) is:
\[ x^2(-p) + 4pxy + 2y^2p^2 = 0 \]