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 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 line \( x + y = 5 \) can be rewritten in slope-intercept form as \( y = -x + 5 \). The lines represented by \( |x| = |y| \) correspond to the equations \( y = x \) and \( y = -x \).

### Step 2: Choose a point on the line \( x + y = 5 \)
Let’s denote a point \( S \) on the line \( x + y = 5 \) as \( (h, 5 - h) \), where \( h \) is a variable representing the x-coordinate.

### Step 3: Calculate the distance from point \( S \) to the lines \( y = x \) and \( y = -x \)
Using the formula for the distance from a point \( (x_1, y_1) \) to a line \( Ax + By + C = 0 \):
\[ \text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
For the line \( y = x \) (which can be rewritten as \( x - y = 0 \)):
- Here, \( A = 1, B = -1, C = 0 \)
- The distance from \( S(h, 5 - h) \) to this line is:
\[ \text{Distance to } y = x = \frac{|h - (5 - h)|}{\sqrt{1^2 + (-1)^2}} = \frac{|2h - 5|}{\sqrt{2}} \]

For the line \( y = -x \) (which can be rewritten as \( x + y = 0 \)):
- Here, \( A = 1, B = 1, C = 0 \)
- The distance from \( S(h, 5 - h) \) to this line is:
\[ \text{Distance to } y = -x = \frac{|h + (5 - h)|}{\sqrt{1^2 + 1^2}} = \frac{|5|}{\sqrt{2}} = \frac{5}{\sqrt{2}} \]

### Step 4: Set the distances equal
Since we want the distances to be equal:
\[ \frac{|2h - 5|}{\sqrt{2}} = \frac{5}{\sqrt{2}} \]
This simplifies to:
\[ |2h - 5| = 5 \]

### Step 5: Solve the absolute value equation
This gives us two cases to consider:
1. \( 2h - 5 = 5 \)
2. \( 2h - 5 = -5 \)

**Case 1:**
\[ 2h - 5 = 5 \implies 2h = 10 \implies h = 5 \]
Thus, the point is \( (5, 0) \).

**Case 2:**
\[ 2h - 5 = -5 \implies 2h = 0 \implies h = 0 \]
Thus, the point is \( (0, 5) \).

### Step 6: Conclusion
The coordinates of the points on the line \( x + y = 5 \) that are equidistant from the lines \( |x| = |y| \) are:
- \( (5, 0) \)
- \( (0, 5) \)