Find the equation of a straight line passing through the point `(-5,4)` and which cuts off an intercept of `sqrt(2)` units between the lines `x+y+1=0` and `x+y-1=0.`

Explanation

To find the equation of a straight line passing through the point \((-5, 4)\) and cutting off an intercept of \(\sqrt{2}\) units between the lines \(x + y + 1 = 0\) and \(x + y - 1 = 0\), we can follow these steps:

### Step 1: Identify the distance between the two parallel lines
The lines \(x + y + 1 = 0\) and \(x + y - 1 = 0\) are parallel. The distance \(d\) between two parallel lines of the form \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is given by the formula:

\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]

For our lines:
- \(C_1 = 1\) (from \(x + y + 1 = 0\))
- \(C_2 = -1\) (from \(x + y - 1 = 0\))

Thus, the distance \(d\) is:

\[ d = \frac{|-1 - 1|}{\sqrt{1^2 + 1^2}} = \frac{|-2|}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

This confirms that the distance between the two lines is indeed \(\sqrt{2}\).

### Step 2: Determine the equation of the required line
The required line must be perpendicular to the given lines. The slope of the lines \(x + y + 1 = 0\) and \(x + y - 1 = 0\) is \(-1\). Therefore, the slope of the line we are looking for, which is perpendicular, will be \(1\).

We can write the equation of the line in point-slope form:

\[ y - y_1 = m(x - x_1) \]

Substituting the point \((-5, 4)\) and the slope \(m = 1\):

\[ y - 4 = 1(x + 5) \]

This simplifies to:

\[ y - 4 = x + 5 \]

Rearranging gives:

\[ x - y + 9 = 0 \]

### Step 3: Final equation of the line
Thus, the equation of the required line is:

\[ x - y + 9 = 0 \]