Find the equation of a line with slope –1 and whose perpendicular distance from the origin is equal to 5.

Answer

To find the equation of a line with a slope of -1 and a perpendicular distance of 5 from the origin, we can follow these steps:

### Step 1: Understand the equation of a line
The general equation of a line can be expressed as:
\[ Ax + By + C = 0 \]
where \( A \), \( B \), and \( C \) are constants.

### Step 2: Set the slope
Given that the slope \( m \) of the line is -1, we can express the line in slope-intercept form:
\[ y = -x + c \]
This can be rearranged to:
\[ x + y - c = 0 \]
Here, \( A = 1 \), \( B = 1 \), and \( C = -c \).

### Step 3: Use the formula for perpendicular distance
The formula for the perpendicular distance \( d \) from a point \( (x_1, y_1) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
In our case, the point is the origin \( (0, 0) \), so \( x_1 = 0 \) and \( y_1 = 0 \). The distance from the origin to the line is given as 5.

### Step 4: Substitute values into the distance formula
Substituting the values into the distance formula:
\[ 5 = \frac{|1(0) + 1(0) - c|}{\sqrt{1^2 + 1^2}} \]
This simplifies to:
\[ 5 = \frac{| -c |}{\sqrt{2}} \]

### Step 5: Solve for \( c \)
To eliminate the fraction, multiply both sides by \( \sqrt{2} \):
\[ 5\sqrt{2} = | -c | \]
This implies:
\[ |c| = 5\sqrt{2} \]

### Step 6: Determine the values of \( c \)
From the absolute value equation, we have two cases:
1. \( -c = 5\sqrt{2} \) → \( c = -5\sqrt{2} \)
2. \( -c = -5\sqrt{2} \) → \( c = 5\sqrt{2} \)

### Step 7: Write the equations of the lines
Substituting these values of \( c \) back into the line equation \( x + y - c = 0 \):
1. For \( c = -5\sqrt{2} \):
\[ x + y + 5\sqrt{2} = 0 \]
2. For \( c = 5\sqrt{2} \):
\[ x + y - 5\sqrt{2} = 0 \]

### Final Answer
Thus, the equations of the lines are:
1. \( x + y + 5\sqrt{2} = 0 \)
2. \( x + y - 5\sqrt{2} = 0 \)