Find the co-ordinates of the points on the curve `y= (x)/( 1-x^(2)) " for which " (dy)/( dx)= 1 `

Answer

To find the coordinates of the points on the curve \( y = \frac{x}{1 - x^2} \) for which \( \frac{dy}{dx} = 1 \), we will follow these steps:

### Step 1: Differentiate \( y \)
Given:
\[ y = \frac{x}{1 - x^2} \]
We will use the quotient rule to differentiate \( y \). The quotient rule states that if \( y = \frac{u}{v} \), then:
\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]
Here, \( u = x \) and \( v = 1 - x^2 \).

Calculating \( \frac{du}{dx} \) and \( \frac{dv}{dx} \):
- \( \frac{du}{dx} = 1 \)
- \( \frac{dv}{dx} = -2x \)

Now applying the quotient rule:
\[ \frac{dy}{dx} = \frac{(1 - x^2)(1) - (x)(-2x)}{(1 - x^2)^2} \]
\[ = \frac{1 - x^2 + 2x^2}{(1 - x^2)^2} \]
\[ = \frac{1 + x^2}{(1 - x^2)^2} \]

### Step 2: Set \( \frac{dy}{dx} = 1 \)
Now, we set the derivative equal to 1:
\[ \frac{1 + x^2}{(1 - x^2)^2} = 1 \]

### Step 3: Solve the equation
Cross-multiplying gives:
\[ 1 + x^2 = (1 - x^2)^2 \]
Expanding the right side:
\[ 1 + x^2 = 1 - 2x^2 + x^4 \]
Rearranging terms:
\[ x^4 - 3x^2 = 0 \]
Factoring out \( x^2 \):
\[ x^2(x^2 - 3) = 0 \]
This gives us:
\[ x^2 = 0 \quad \text{or} \quad x^2 = 3 \]
Thus:
\[ x = 0 \quad \text{or} \quad x = \pm \sqrt{3} \]

### Step 4: Find corresponding \( y \) values
Now we will find the corresponding \( y \) values for each \( x \).

1. **For \( x = 0 \)**:
\[ y = \frac{0}{1 - 0^2} = 0 \]
So, the point is \( (0, 0) \).

2. **For \( x = \sqrt{3} \)**:
\[ y = \frac{\sqrt{3}}{1 - (\sqrt{3})^2} = \frac{\sqrt{3}}{1 - 3} = \frac{\sqrt{3}}{-2} = -\frac{\sqrt{3}}{2} \]
So, the point is \( \left(\sqrt{3}, -\frac{\sqrt{3}}{2}\right) \).

3. **For \( x = -\sqrt{3} \)**:
\[ y = \frac{-\sqrt{3}}{1 - (-\sqrt{3})^2} = \frac{-\sqrt{3}}{1 - 3} = \frac{-\sqrt{3}}{-2} = \frac{\sqrt{3}}{2} \]
So, the point is \( \left(-\sqrt{3}, \frac{\sqrt{3}}{2}\right) \).

### Final Points
The coordinates of the points on the curve where \( \frac{dy}{dx} = 1 \) are:
1. \( (0, 0) \)
2. \( \left(\sqrt{3}, -\frac{\sqrt{3}}{2}\right) \)
3. \( \left(-\sqrt{3}, \frac{\sqrt{3}}{2}\right) \)