The differential equation of the rectangular hyperbola whose axes are the asymptotes of the hyperbola, is

Options

A. `y(dy)/(dx)=x`
B. `x(dy)/(dx)=-y`
C. `x(dy)/(dx)=y`
D. `x dy +ydx=C`

Explanation

To find the differential equation of the rectangular hyperbola whose axes are the asymptotes, we can follow these steps:

### Step 1: Understand the equation of the rectangular hyperbola
The standard equation of a rectangular hyperbola is given by:
\[ xy = c \]
where \(c\) is a constant.

### Step 2: Differentiate the equation
To find the differential equation, we need to differentiate the equation \(xy = c\) with respect to \(x\). We will use the product rule for differentiation.

The product rule states that if you have two functions \(u\) and \(v\), then:
\[ \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx} \]

In our case, let \(u = x\) and \(v = y\). Thus, we differentiate:
\[ \frac{d}{dx}(xy) = \frac{d}{dx}(c) \]

### Step 3: Apply the product rule
Applying the product rule:
\[ x \frac{dy}{dx} + y \frac{dx}{dx} = 0 \]
Since \(\frac{dx}{dx} = 1\), we can simplify this to:
\[ x \frac{dy}{dx} + y = 0 \]

### Step 4: Rearranging the equation
Now, we can rearrange the equation to express \(\frac{dy}{dx}\):
\[ x \frac{dy}{dx} = -y \]
\[ \frac{dy}{dx} = -\frac{y}{x} \]

### Step 5: Conclusion
Thus, the required differential equation of the rectangular hyperbola whose axes are the asymptotes is:
\[ \frac{dy}{dx} = -\frac{y}{x} \]