Consider the curve C satisfying `2xydy = (x^(2) + y^(2) + 1) dx`
Statement 1 : Curve C represents a family of rectangular hyperbola with centre on x- axis and Statement 2 : Curve C represents a family of rectangular hyperbolas with centre on y - axis

Options

A. Statement - 1 is True, Statement - 2 is True, Statement - 2 is a correct explanation for Statement - 2
B. Statement - 1 is True, Statement - 2 is True, Statement - 2 is NOT a correct explanation for Statement - 2
C. Statement - 1 is True , Statement - 2 is False
D. Statement - 1 is False, Statement - 2 is True

Answer

To solve the given problem, we need to analyze the differential equation and determine the nature of the curve it represents. Let's go through the solution step by step.

### Step 1: Write the given differential equation
The differential equation is given by:
\[ 2xy \, dy = (x^2 + y^2 + 1) \, dx \]

### Step 2: Rearrange the equation
We can rearrange the equation to isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{x^2 + y^2 + 1}{2xy} \]

### Step 3: Substitute \( y^2 \) with \( b \)
Let’s substitute \( y^2 \) with \( b \):
\[ b = y^2 \]
Thus, \( dy = \frac{1}{2\sqrt{b}} db \).

### Step 4: Rewrite the equation using the substitution
Substituting \( dy \) into the equation gives:
\[ \frac{1}{2\sqrt{b}} \frac{db}{dx} = \frac{x^2 + b + 1}{2x\sqrt{b}} \]

### Step 5: Simplify the equation
Multiplying through by \( 2x\sqrt{b} \) to eliminate the denominators results in:
\[ x \, db = (x^2 + b + 1) \, dx \]

### Step 6: Rearranging terms
Rearranging gives:
\[ db - \frac{b}{x^2 + 1} \, dx = dx \]

### Step 7: Separate variables
This can be separated as:
\[ \frac{db}{b} = \frac{dx}{x} + \frac{dx}{x^2 + 1} \]

### Step 8: Integrate both sides
Integrating both sides leads to:
\[ \ln |b| = \ln |x| + \tan^{-1}(x) + C \]

### Step 9: Exponentiate to solve for \( b \)
Exponentiating gives:
\[ b = k \cdot x \cdot e^{\tan^{-1}(x)} \]
where \( k = e^C \).

### Step 10: Substitute back for \( b \)
Substituting back for \( b = y^2 \):
\[ y^2 = k \cdot x \cdot e^{\tan^{-1}(x)} \]

### Step 11: Analyze the resulting equation
The resulting equation represents a family of curves. To determine if they are hyperbolas, we analyze the behavior of the equation.

### Conclusion
The curves represented by the equation are hyperbolas centered on the x-axis, confirming that Statement 1 is true and Statement 2 is false.

### Final Answer
Thus, the correct answer is that Statement 1 is true and Statement 2 is false.

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