The coordinates of the points (s) on the line x + y = 5 , which is/ are equidistant from the lines `|x|=|y|` , is/ are

Options

A. (5,0)
B. (0,5)
C. (-5,0)
D. (0,-5)

Answer

To solve the problem of finding the coordinates of the points on the line \( x + y = 5 \) that are equidistant from the lines \( |x| = |y| \), we can follow these steps:

### Step 1: Understand the lines involved
The equation \( |x| = |y| \) represents the lines:
1. \( y = x \) (first quadrant)
2. \( y = -x \) (second and fourth quadrants)

These lines intersect at the origin (0,0) and form a "V" shape.

### Step 2: Find the equation of the line
We are given the line \( x + y = 5 \). We can rewrite it in slope-intercept form:
\[ y = -x + 5 \]

### Step 3: Determine the distance from a point to the lines
To find points on the line \( x + y = 5 \) that are equidistant from the lines \( y = x \) and \( y = -x \), we need to calculate the perpendicular distance from a point \( (x_0, y_0) \) on the line to these two lines.

The 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 the line \( y = x \) (or \( x - y = 0 \)):
- \( A = 1, B = -1, C = 0 \)
- Distance from point \( (x_0, y_0) \):
\[ d_1 = \frac{|x_0 - y_0|}{\sqrt{1^2 + (-1)^2}} = \frac{|x_0 - y_0|}{\sqrt{2}} \]

For the line \( y = -x \) (or \( x + y = 0 \)):
- \( A = 1, B = 1, C = 0 \)
- Distance from point \( (x_0, y_0) \):
\[ d_2 = \frac{|x_0 + y_0|}{\sqrt{1^2 + 1^2}} = \frac{|x_0 + y_0|}{\sqrt{2}} \]

### Step 4: Set the distances equal
For the point to be equidistant from both lines:
\[ d_1 = d_2 \]
This gives us:
\[ \frac{|x_0 - y_0|}{\sqrt{2}} = \frac{|x_0 + y_0|}{\sqrt{2}} \]

Removing \( \sqrt{2} \) from both sides:
\[ |x_0 - y_0| = |x_0 + y_0| \]

### Step 5: Analyze the absolute value equation
This absolute value equation can be split into two cases:
1. \( x_0 - y_0 = x_0 + y_0 \) → \( -y_0 = y_0 \) → \( y_0 = 0 \)
2. \( x_0 - y_0 = -(x_0 + y_0) \) → \( 2x_0 = 0 \) → \( x_0 = 0 \)

### Step 6: Substitute back into the line equation
From the first case \( y_0 = 0 \):
- Substitute into \( x + y = 5 \):
\[ x + 0 = 5 \Rightarrow x = 5 \]
So one point is \( (5, 0) \).

From the second case \( x_0 = 0 \):
- Substitute into \( x + y = 5 \):
\[ 0 + y = 5 \Rightarrow y = 5 \]
So another point is \( (0, 5) \).

### Step 7: Conclusion
The coordinates of the points on the line \( x + y = 5 \) that are equidistant from the lines \( |x| = |y| \) are:
- \( (5, 0) \)
- \( (0, 5) \)