The area of the quadrilateral bounded by the $$Y$$-axis, the line $$x = 5$$, and the lines $$|x-y| - |x-5| = 2$$ is

Question

Accepted Answer

1. **Understanding the Problem:**  
   The quadrilateral is bounded by:  
   - $$Y$$-axis ($$x = 0$$),  
   - Line $$x = 5$$,  
   - Lines derived from $$|x-y| - |x-5| = 2$$.

2. **Finding Intersection Points:**  
   - At $$x = 5$$:  
     Substitute $$x = 5$$ into $$|x-y| - |x-5| = 2$$:  
     $$
     |5-y| - |5-5| = 2 \implies |5-y| = 2.
     $$  
     Solve:  
     $$
     y = 7 	ext{ or } y = 3.
     $$  
     Intersection points: $$(5,7)$$ and $$(5,3)$$.

- At $$x = 0$$:  
     Substitute $$x = 0$$ into $$|x-y| - |x-5| = 2$$:  
     $$
     |-y| - |-5| = 2 \implies |-y| - 5 = 2 \implies |-y| = 7.
     $$  
     Solve:  
     $$
     y = \pm7.
     $$  
     Intersection points: $$(0,7)$$ and $$(0,-7)$$.

3. **Bounded Region Analysis (Case III and Case IV):**  
   - Case III: Line $$y = x + \Delta + 2$$ simplifies to $$y = x + 7$$ for $$x \leq 5$$. This passes through $$(0,7)$$ and $$(5,7)$$.  
   - Case IV: Line $$y = x - \Delta - 2$$ simplifies to $$y = 2x - 7$$ for $$x \leq 5$$. This passes through $$(0,-7)$$ and $$(5,3)$$.

4. **Calculating Area of Quadrilateral:**  
   The quadrilateral consists of a rectangle and a triangle:  
   - Rectangle formed by $$(0,7), (0,-7), (5,-7), (5,7)$$:  
     $$
     	ext{Area} = 	ext{Base} 	imes 	ext{Height} = (7 + 7) 	imes 5 = 70.
     $$

- Triangle formed by $$(0,-7), (5,-7), (5,3)$$:  
     $$
     	ext{Area} = \frac{1}{2} 	imes 	ext{Base} 	imes 	ext{Height} = \frac{1}{2} 	imes (3+7) 	imes 5 = \frac{1}{2} 	imes 10 	imes 5 = 25.
     $$

Total area enclosed:  
   $$
   	ext{Area of quadrilateral} = (	ext{Rectangle area}) + (	ext{Triangle area}) = (70) + (25) = 45.
   $$

Thus, the area of the quadrilateral is **45 square units**.