What are the degree and order respectively of differential equation of the family of rectangular hyperbolas whose axis of symmetry are the coordinate axis ?

Options

A. 1,1
B. 1,2
C. 2,1
D. 2,2

Explanation

To find the degree and order of the differential equation of the family of rectangular hyperbolas whose axes of symmetry are the coordinate axes, we can follow these steps:

### Step 1: Understand the Equation of Rectangular Hyperbola
The general equation of a rectangular hyperbola centered at the origin with axes along the coordinate axes is given by:
\[ xy = c^2 \]
where \(c\) is a constant.

### Step 2: Differentiate the Equation
To find the differential equation, we need to eliminate the constant \(c\). We can differentiate the equation implicitly with respect to \(x\):
\[ \frac{d}{dx}(xy) = \frac{d}{dx}(c^2) \]
Using the product rule on the left side, we get:
\[ y + x\frac{dy}{dx} = 0 \]

### Step 3: Rearranging the Equation
Now, we can rearrange this equation to express \(\frac{dy}{dx}\):
\[ x\frac{dy}{dx} = -y \]
\[ \frac{dy}{dx} = -\frac{y}{x} \]

### Step 4: Forming the Differential Equation
The equation \(\frac{dy}{dx} = -\frac{y}{x}\) is a first-order differential equation.

### Step 5: Identify the Order and Degree
The order of a differential equation is the highest derivative present in the equation. Here, the highest derivative is \(\frac{dy}{dx}\), which is of order 1.

The degree of a differential equation is the power of the highest derivative when the equation is a polynomial in derivatives. Since \(\frac{dy}{dx}\) is to the power of 1, the degree is also 1.

### Final Answer
Thus, the degree and order of the differential equation of the family of rectangular hyperbolas whose axes of symmetry are the coordinate axes are:
- **Order**: 1
- **Degree**: 1