The equation of the line passing through the point (-3,1) and bisecting the angle between coordinate axes is

Options

A. x+y+2=0
B. `-x+y+2=0`
C. `x-y+4=0`
D. `2x+y+5=0`

Explanation

To find the equation of the line passing through the point (-3, 1) and bisecting the angle between the coordinate axes, we can follow these steps:

### Step 1: Understand the angle bisector
The angle bisector of the coordinate axes is the line that divides the angle between the x-axis and y-axis into two equal angles. The angle between the axes is 90 degrees, so the bisector will make an angle of 45 degrees with each axis.

### Step 2: Determine the slope of the line
The slope \( m \) of a line that makes an angle \( \theta \) with the positive x-axis is given by \( m = \tan(\theta) \). Since the line bisects the angle between the axes, the angle \( \theta \) will be 135 degrees (90 degrees + 45 degrees).

Calculating the slope:
\[ m = \tan(135^\circ) = \tan(180^\circ - 45^\circ) = -\tan(45^\circ) = -1 \]

### Step 3: Use the point-slope form of the equation of a line
The point-slope form of the equation of a line is given by:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is the point through which the line passes. Here, \( (x_1, y_1) = (-3, 1) \) and \( m = -1 \).

Substituting the values into the equation:
\[ y - 1 = -1(x - (-3)) \]
This simplifies to:
\[ y - 1 = -1(x + 3) \]

### Step 4: Simplify the equation
Distributing the -1:
\[ y - 1 = -x - 3 \]
Now, add 1 to both sides:
\[ y = -x - 2 \]

### Step 5: Rearranging to standard form
To express this in standard form \( Ax + By + C = 0 \):
\[ x + y + 2 = 0 \]

Thus, the equation of the line is:
\[ \boxed{x + y + 2 = 0} \]