The points on the curve `y =x/((1-x^2))` where the tangent is inclined at angle `pi//4` to the x-axis

Options

A. `(0,0),(sqrt3,-sqrt3//2) `
B. `(0,0),(-sqrt3,sqrt3//2)`
C. `(0, 0), (sqrt3,, sqrt3// 2)`
D. None of these

Answer

To solve the problem of finding the points on the curve \( y = \frac{x}{1 - x^2} \) where the tangent is inclined at an angle of \( \frac{\pi}{4} \) to the x-axis, we will follow these steps:

### Step 1: Determine the slope of the tangent
The slope \( m \) of the tangent line at an angle \( \theta \) with the x-axis is given by:
\[ m = \tan(\theta) \]
For \( \theta = \frac{\pi}{4} \):
\[ m = \tan\left(\frac{\pi}{4}\right) = 1 \]

### Step 2: Differentiate the curve
We need to find the derivative \( \frac{dy}{dx} \) of the curve \( y = \frac{x}{1 - x^2} \). We will use the quotient rule for differentiation:
\[ \frac{dy}{dx} = \frac{(1 - x^2)(1) - x(0 - 2x)}{(1 - x^2)^2} \]
Simplifying this:
\[ \frac{dy}{dx} = \frac{(1 - x^2) + 2x^2}{(1 - x^2)^2} = \frac{1 + x^2}{(1 - x^2)^2} \]

### Step 3: Set the derivative equal to the slope
We set the derivative equal to the slope we found in Step 1:
\[ \frac{1 + x^2}{(1 - x^2)^2} = 1 \]

### Step 4: Cross-multiply and simplify
Cross-multiplying gives:
\[ 1 + x^2 = (1 - x^2)^2 \]
Expanding the right side:
\[ 1 + x^2 = 1 - 2x^2 + x^4 \]
Rearranging the equation:
\[ x^4 - 3x^2 = 0 \]

### Step 5: Factor the equation
Factoring out \( x^2 \):
\[ x^2(x^2 - 3) = 0 \]
This gives us:
\[ x^2 = 0 \quad \text{or} \quad x^2 = 3 \]
Thus, the solutions for \( x \) are:
\[ x = 0, \quad x = \sqrt{3}, \quad x = -\sqrt{3} \]

### Step 6: Find corresponding \( y \) values
Now we will find the corresponding \( y \) values for each \( x \):
1. For \( x = 0 \):
\[ y = \frac{0}{1 - 0^2} = 0 \quad \Rightarrow \quad (0, 0) \]
2. For \( x = \sqrt{3} \):
\[ y = \frac{\sqrt{3}}{1 - 3} = \frac{\sqrt{3}}{-2} = -\frac{\sqrt{3}}{2} \quad \Rightarrow \quad \left(\sqrt{3}, -\frac{\sqrt{3}}{2}\right) \]
3. For \( x = -\sqrt{3} \):
\[ y = \frac{-\sqrt{3}}{1 - 3} = \frac{-\sqrt{3}}{-2} = \frac{\sqrt{3}}{2} \quad \Rightarrow \quad \left(-\sqrt{3}, \frac{\sqrt{3}}{2}\right) \]

### Final Points
The points on the curve where the tangent is inclined at an angle of \( \frac{\pi}{4} \) to the x-axis are:
1. \( (0, 0) \)
2. \( \left(\sqrt{3}, -\frac{\sqrt{3}}{2}\right) \)
3. \( \left(-\sqrt{3}, \frac{\sqrt{3}}{2}\right) \)