The vertices of a `triangleOBC` are `O(0,0) , B(-3,-1), C(-1,-3)`. Find the equation of the line parallel to BC and intersecting the sides OB and OC and whose perpendicular distance from the origin is `1/2`.

Options

A. ` x + y + (1)/(2) = 0`
B. `x + y - (1)/(2) = 0`
C. `x + y - (1)/(sqrt2) = 0`
D. `x + y + (1)/(sqrt2) = 0`

Answer

To solve the problem step by step, we will follow the outlined approach to find the equation of the line parallel to BC, intersecting the sides OB and OC, and having a perpendicular distance of \( \frac{1}{2} \) from the origin.

### Step 1: Find the equation of line BC

The coordinates of points B and C are:
- \( B(-3, -1) \)
- \( C(-1, -3) \)

To find the slope of line BC, we use the formula:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - (-1)}{-1 - (-3)} = \frac{-2}{2} = -1 \]

Now, using the point-slope form of the equation of a line, we can write the equation of line BC:
\[ y - y_1 = m(x - x_1) \]
Substituting \( m = -1 \), \( (x_1, y_1) = (-3, -1) \):
\[ y + 1 = -1(x + 3) \]
Simplifying this:
\[ y + 1 = -x - 3 \implies x + y + 4 = 0 \]

### Step 2: Write the equation of the line parallel to BC

The equation of line BC is \( x + y + 4 = 0 \). A line parallel to this will have the same coefficients for \( x \) and \( y \), so we can write:
\[ x + y + \lambda = 0 \]
where \( \lambda \) is a constant we need to determine.

### Step 3: Find the perpendicular distance from the origin to the line

The formula for the perpendicular distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For our line \( x + y + \lambda = 0 \), we have \( A = 1 \), \( B = 1 \), and \( C = \lambda \). The origin has coordinates \( (0, 0) \).

Substituting into the distance formula:
\[ d = \frac{|1(0) + 1(0) + \lambda|}{\sqrt{1^2 + 1^2}} = \frac{|\lambda|}{\sqrt{2}} \]
We know the distance \( d \) is \( \frac{1}{2} \):
\[ \frac{|\lambda|}{\sqrt{2}} = \frac{1}{2} \]
Multiplying both sides by \( \sqrt{2} \):
\[ |\lambda| = \frac{\sqrt{2}}{2} \]

### Step 4: Determine the values of \( \lambda \)

This gives us two possible values for \( \lambda \):
\[ \lambda = \frac{\sqrt{2}}{2} \quad \text{or} \quad \lambda = -\frac{\sqrt{2}}{2} \]

### Step 5: Determine which value of \( \lambda \) is valid

We need to check which value of \( \lambda \) will allow the line to intersect the sides OB and OC.

1. For \( \lambda = \frac{\sqrt{2}}{2} \):
\[ x + y + \frac{\sqrt{2}}{2} = 0 \]
This line is above the origin.

2. For \( \lambda = -\frac{\sqrt{2}}{2} \):
\[ x + y - \frac{\sqrt{2}}{2} = 0 \]
This line is below the origin.

Since the line must intersect both OB and OC, we will choose \( \lambda = -\frac{\sqrt{2}}{2} \).

### Step 6: Write the final equation of the required line

Substituting \( \lambda = -\frac{\sqrt{2}}{2} \) into the equation:
\[ x + y - \frac{\sqrt{2}}{2} = 0 \]
Multiplying through by 2 to eliminate the fraction:
\[ 2x + 2y - \sqrt{2} = 0 \]

Thus, the final equation of the required line is:
\[ 2x + 2y - \sqrt{2} = 0 \]