The equation of the pair of straight lines, each of which makes an angle `alpha` with the line `y=x` is

Options

A. `x^(2)+2xy sec 2 alpha+y^(2)=0`
B. `x^(2)+2xy cosec 2 alpha+y^(2)=0`
C. `x^(2)-2xy cosec 2 alpha +y^(2)=0`
D. `x^(2)-2xy sec 2 alpha +y^(2)=0`

Explanation

To find the equation of the pair of straight lines that each make an angle \( \alpha \) with the line \( y = x \), we can follow these steps:

### Step 1: Understand the angle with respect to the x-axis
The line \( y = x \) makes an angle of \( 45^\circ \) with the x-axis. If a line makes an angle \( \alpha \) with \( y = x \), then the angles made with the x-axis will be \( 45^\circ - \alpha \) and \( 45^\circ + \alpha \).

### Step 2: Determine the slopes of the lines
The slopes of the lines can be determined using the tangent of the angles:
- For the first line:
\[ m_1 = \tan(45^\circ - \alpha) = \frac{1 - \tan(\alpha)}{1 + \tan(\alpha)} \]
- For the second line:
\[ m_2 = \tan(45^\circ + \alpha) = \frac{1 + \tan(\alpha)}{1 - \tan(\alpha)} \]

### Step 3: Write the equations of the lines
Using the point-slope form of the line equation \( y - y_1 = m(x - x_1) \), and since both lines pass through the origin (0,0), we can write:
- For the first line:
\[ y = m_1 x = \tan(45^\circ - \alpha) x \]
- For the second line:
\[ y = m_2 x = \tan(45^\circ + \alpha) x \]

### Step 4: Formulate the equation of the pair of lines
The general equation of a pair of straight lines through the origin can be expressed as:
\[ y^2 = m_1 x^2 + m_2 x^2 \]
Thus, the equation can be written as:
\[ y^2 - (m_1 + m_2)xy + m_1 m_2 x^2 = 0 \]

### Step 5: Substitute the values of \( m_1 \) and \( m_2 \)
Substituting the values of \( m_1 \) and \( m_2 \) into the equation gives:
\[ y^2 - (m_1 + m_2)xy + m_1 m_2 x^2 = 0 \]

### Step 6: Simplify the equation
Using the identities for \( m_1 \) and \( m_2 \):
- \( m_1 + m_2 = \tan(45^\circ - \alpha) + \tan(45^\circ + \alpha) \)
- \( m_1 m_2 = \tan(45^\circ - \alpha) \tan(45^\circ + \alpha) \)

This leads to the final equation of the pair of lines.

### Final Result
The equation of the pair of straight lines that each make an angle \( \alpha \) with the line \( y = x \) is:
\[ y^2 - 2xy \cos(2\alpha) + x^2 = 0 \]