The equations of the lines through `(-1,-1)` and making angle `45^@` with the line `x+y=0` are given by

Options

A. `x^(2) -xy + x - y =0`
B. `xy -y^(2) + x - y = 0`
C. `xy + xy+ y=0`
D. `xy + x+ y +1=0`

Explanation

To find the equations of the lines through the point \((-1, -1)\) that make an angle of \(45^\circ\) with the line \(x + y = 0\), follow these steps:

### Step 1: Determine the slope of the line \(x + y = 0\)
The equation \(x + y = 0\) can be rewritten in slope-intercept form as:
\[ y = -x \]
From this, we can see that the slope \(m_1\) of the line is:
\[ m_1 = -1 \]