The differential equation of rectangular hyperbolas whose axes are asymptotes of the hyperbola ` x^2 - y^2= a^2`, is :

Options

A. `y (dy)/(dx) =x`
B. `x (dy)/(dx)=-y`
C. `x(dy)/(dx)=y`
D. `xdy +ydx=3`

Answer

To find the differential equation of rectangular hyperbolas whose axes are asymptotes of the hyperbola \( x^2 - y^2 = a^2 \), we will follow these steps:

### Step 1: Differentiate the given equation
We start with the equation of the hyperbola:
\[ x^2 - y^2 = a^2 \]
We differentiate both sides with respect to \( x \).

### Step 2: Apply the differentiation
Using implicit differentiation:
\[ \frac{d}{dx}(x^2) - \frac{d}{dx}(y^2) = \frac{d}{dx}(a^2) \]
This gives us:
\[ 2x - 2y \frac{dy}{dx} = 0 \]

### Step 3: Simplify the equation
We can simplify this equation by factoring out the common term:
\[ 2x - 2y \frac{dy}{dx} = 0 \implies x - y \frac{dy}{dx} = 0 \]

### Step 4: Rearranging the equation
Rearranging gives us:
\[ y \frac{dy}{dx} = x \]

### Final Result
Thus, the differential equation of rectangular hyperbolas whose axes are asymptotes of the hyperbola \( x^2 - y^2 = a^2 \) is:
\[ y \frac{dy}{dx} = x \]