Family of curves whose tangent at a point with its intersection with the curve `xy = c^2` form an angle of `pi/4` is

Options

A. `y^(2)-2xy-x^(2)=k`
B. `y^(2)+2xy-x^(2)=k`
C. `y=x-2ctan^(-1)((x)/(c))+k`
D. `y=cIn|(c+x)/(c-x)|-x+k`

Answer

To find the family of curves whose tangent at a point with its intersection with the curve \(xy = c^2\) forms an angle of \(\frac{\pi}{4}\), we can follow these steps:

### Step 1: Understand the curves
We have two curves:
1. The curve \(xy = c^2\)
2. The curve \(y = f(x)\)

### Step 2: Find the slope of the tangent to the first curve
From the equation \(xy = c^2\), we can express \(y\) in terms of \(x\):
\[ y = \frac{c^2}{x} \]
Now, differentiate this with respect to \(x\) to find the slope \(m_1\):
\[ \frac{dy}{dx} = -\frac{c^2}{x^2} \]
Thus, the slope of the tangent to the curve \(xy = c^2\) at the point \((x, y)\) is:
\[ m_1 = -\frac{c^2}{x^2} \]

### Step 3: Find the slope of the tangent to the second curve
For the curve \(y = f(x)\), the slope of the tangent is given by:
\[ m_2 = f'(x) \]

### Step 4: Use the angle condition
We know that the angle between the tangents of the two curves is \(\frac{\pi}{4}\). The tangent of the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is given by:
\[ \tan(\theta) = \frac{m_1 - m_2}{1 + m_1 m_2} \]
Since \(\theta = \frac{\pi}{4}\), we have:
\[ \tan\left(\frac{\pi}{4}\right) = 1 \]
Thus, we can set up the equation:
\[ 1 = \frac{m_1 - m_2}{1 + m_1 m_2} \]

### Step 5: Substitute the slopes into the equation
Substituting \(m_1\) and \(m_2\) into the equation:
\[ 1 = \frac{-\frac{c^2}{x^2} - f'(x)}{1 - \frac{c^2}{x^2} f'(x)} \]

### Step 6: Cross-multiply and simplify
Cross-multiplying gives:
\[ 1 - \frac{c^2}{x^2} f'(x) = -\frac{c^2}{x^2} - f'(x) \]
Rearranging this, we have:
\[ 1 + \frac{c^2}{x^2} = f'(x) + \frac{c^2}{x^2} f'(x) \]
Factoring out \(f'(x)\):
\[ 1 + \frac{c^2}{x^2} = f'(x) \left(1 + \frac{c^2}{x^2}\right) \]

### Step 7: Solve for \(f'(x)\)
Assuming \(1 + \frac{c^2}{x^2} \neq 0\), we can divide both sides:
\[ f'(x) = 1 \]

### Step 8: Integrate to find \(f(x)\)
Integrating \(f'(x)\):
\[ f(x) = x + k \]
where \(k\) is the constant of integration.

### Step 9: Write the family of curves
Thus, the family of curves is given by:
\[ y = x + k \]