STATEMENT-1 : The differential equation of all circles in a plane can be of order 3.
and
STATEMENT-2 : General equation of a circle in plane has three independent constant parameters

Options

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

Answer

To solve the problem, we need to analyze both statements regarding circles in a plane and their differential equations.

### Step 1: Write the General Equation of a Circle
The general equation of a circle in a plane is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where \( g \), \( f \), and \( c \) are constants.

### Step 2: Identify the Parameters
From the general equation, we can see that there are three independent constant parameters:
- \( g \) (related to the x-coordinate of the center)
- \( f \) (related to the y-coordinate of the center)
- \( c \) (related to the radius of the circle)

### Step 3: Differentiate the Equation
To find the differential equation that represents all circles, we need to differentiate the general equation. Since we have three parameters, we will differentiate the equation three times to eliminate these parameters.

1. **First Derivative**:
\[ \frac{d}{dx}(x^2 + y^2 + 2gx + 2fy + c) = 0 \]
This gives us:
\[ 2x + 2y\frac{dy}{dx} + 2g + 2f\frac{dy}{dx} = 0 \]

2. **Second Derivative**:
We differentiate the first derivative again to eliminate \( g \) and \( f \):
\[ \frac{d}{dx}(2x + 2y\frac{dy}{dx} + 2g + 2f\frac{dy}{dx}) = 0 \]

3. **Third Derivative**:
We differentiate the second derivative to eliminate the last parameter:
\[ \frac{d}{dx}(\text{Second Derivative}) = 0 \]

### Step 4: Order of the Differential Equation
After differentiating three times, we will arrive at a differential equation that does not contain any of the parameters \( g \), \( f \), or \( c \). Thus, the order of the differential equation representing all circles in a plane is 3.

### Conclusion
- **Statement 1**: The differential equation of all circles in a plane can be of order 3. (True)
- **Statement 2**: The general equation of a circle in a plane has three independent constant parameters. (True)

Since both statements are true, and Statement 2 correctly explains Statement 1, we conclude that both statements are valid.

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