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 the asymptotes of the hyperbola given by the equation \( x^2 - y^2 = a^2 \), we can follow these steps:

### Step 1: Understand the given hyperbola
The equation \( x^2 - y^2 = a^2 \) represents a hyperbola centered at the origin with transverse axis along the x-axis and conjugate axis along the y-axis. The asymptotes of this hyperbola are given by the equations \( y = x \) and \( y = -x \).

### Step 2: Differentiate the hyperbola equation
We differentiate the equation \( x^2 - y^2 = a^2 \) with respect to \( x \):
\[ \frac{d}{dx}(x^2) - \frac{d}{dx}(y^2) = \frac{d}{dx}(a^2) \]
This gives us:
\[ 2x - 2y \frac{dy}{dx} = 0 \]

### Step 3: Simplify the differentiation result
We can simplify the equation obtained from differentiation:
\[ 2x - 2y \frac{dy}{dx} = 0 \]
Dividing through by 2, we have:
\[ x - y \frac{dy}{dx} = 0 \]

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

### Step 5: Final expression
This can be rewritten as:
\[ y \frac{dy}{dx} = x \]
This is the required differential equation of the rectangular hyperbolas whose axes are the asymptotes of the given hyperbola.

### Conclusion
Thus, the differential equation of the rectangular hyperbolas is:
\[ y \frac{dy}{dx} = x \]