Points on the line x + y = 4 which are equidistant from the lines |x| = |y| , are

Options

A. (4,0) , (0,4)
B. (-4,0), (0, -4)
C. (4,0) , (-4, 0)
D. none of these

Answer

To find the points on the line \( x + y = 4 \) that are equidistant from the lines \( |x| = |y| \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Lines**:
The equation \( |x| = |y| \) represents two lines:
- \( y = x \) (first quadrant and third quadrant)
- \( y = -x \) (second quadrant and fourth quadrant)

Therefore, we have the lines:
- Line 1: \( y = x \)
- Line 2: \( y = -x \)

2. **Identify the Line of Interest**:
We need to find points on the line \( x + y = 4 \). We can express this line in slope-intercept form:
\[ y = -x + 4 \]

3. **Determine the Distance from a Point to a Line**:
The distance \( d \) from a point \( (x_0, y_0) \) to a line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

For the lines \( y = x \) and \( y = -x \), we can rewrite them as:
- Line 1: \( x - y = 0 \) (where \( A = 1, B = -1, C = 0 \))
- Line 2: \( x + y = 0 \) (where \( A = 1, B = 1, C = 0 \))

4. **Calculate the Distance from a Point on \( x + y = 4 \)**:
Let’s take a point \( (x, 4 - x) \) on the line \( x + y = 4 \).

**Distance to Line 1**:
\[ d_1 = \frac{|x - (4 - x)|}{\sqrt{1^2 + (-1)^2}} = \frac{|2x - 4|}{\sqrt{2}} = \frac{|2(x - 2)|}{\sqrt{2}} = \sqrt{2}|x - 2| \]

**Distance to Line 2**:
\[ d_2 = \frac{|x + (4 - x)|}{\sqrt{1^2 + 1^2}} = \frac{|4|}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \]

5. **Set the Distances Equal**:
For the points to be equidistant from both lines:
\[ \sqrt{2}|x - 2| = 2\sqrt{2} \]
Dividing both sides by \( \sqrt{2} \):
\[ |x - 2| = 2 \]

6. **Solve for \( x \)**:
This gives us two cases:
- Case 1: \( x - 2 = 2 \) → \( x = 4 \)
- Case 2: \( x - 2 = -2 \) → \( x = 0 \)

7. **Find Corresponding \( y \) Values**:
- For \( x = 4 \):
\[ y = 4 - 4 = 0 \quad \Rightarrow \quad (4, 0) \]
- For \( x = 0 \):
\[ y = 4 - 0 = 4 \quad \Rightarrow \quad (0, 4) \]

### Final Points:
The points on the line \( x + y = 4 \) that are equidistant from the lines \( |x| = |y| \) are:
- \( (4, 0) \)
- \( (0, 4) \)