Find the equation of the lines through `(-2,-1)` and are :
`(i)` parallel to line `x=0`
`(ii)` perpendicular to the line `y=x`.

Answer

To find the equations of the lines through the point \((-2, -1)\) that are (i) parallel to the line \(x = 0\) and (ii) perpendicular to the line \(y = x\), we can follow these steps:

### (i) Line Parallel to \(x = 0\)

1. **Identify the line \(x = 0\)**:
The line \(x = 0\) is the y-axis. This line is vertical.

2. **Determine the slope of the line \(x = 0\)**:
A vertical line has an undefined slope. Therefore, any line parallel to this line will also be vertical.

3. **Write the equation of the vertical line through the point \((-2, -1)\)**:
Since the line is vertical and passes through \((-2, -1)\), its equation will be of the form \(x = -2\).

**Equation of the line parallel to \(x = 0\)**:
\[ \boxed{x = -2} \]

### (ii) Line Perpendicular to \(y = x\)

1. **Identify the slope of the line \(y = x\)**:
The equation \(y = x\) can be rewritten in slope-intercept form \(y = mx + c\) where \(m = 1\). Thus, the slope of the line is \(1\).

2. **Determine the slope of the perpendicular line**:
If two lines are perpendicular, the product of their slopes is \(-1\). Therefore, if the slope of the first line is \(1\), the slope of the line perpendicular to it will be:
\[ m_{\text{perpendicular}} = -\frac{1}{1} = -1 \]

3. **Use the point-slope form to write the equation**:
We can use the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
Here, \((x_1, y_1) = (-2, -1)\) and \(m = -1\). Plugging in these values:
\[ y - (-1) = -1(x - (-2)) \]
Simplifying this:
\[ y + 1 = -1(x + 2) \]
\[ y + 1 = -x - 2 \]
\[ y = -x - 2 - 1 \]
\[ y = -x - 3 \]

**Equation of the line perpendicular to \(y = x\)**:
\[ \boxed{y = -x - 3} \]