Consider two mirrors placed at a right angle to each other and two points A at $ (x,y)$ and B at $ (a,b)$. Suppose a person standing at point A shines a laser pointer so that it hits both mirrors and then hits a person standing at point B (as shown in the picture). What is the total distance that the light ray travels, in terms of $ a$, $ b$, $ x$, and $ y$? Assume that $ x$, $ y$, $ a$, and $ b$ are positive. 

[asy]draw((0,4)--(0,0)--(4,0),linewidth(1));
draw((1,3)--(0,2),MidArrow);
draw((0,2)--(2,0),MidArrow);
draw((2,0)--(3,1),MidArrow);
dot((1,3));
dot((3,1));
label("$A (x,y)$", (1,3),NE);
label("$B (a,b)$", (3,1),NE);[/asy]
1. **Understanding the Problem:**
   We have two mirrors placed at a right angle to each other, forming the x-axis and y-axis. A person at point \( A(x, y) \) shines a laser that reflects off both mirrors and then hits a person at point \( B(a, b) \). We need to find the total distance the light ray travels.

2. **Reflection Concept:**
   When the laser reflects off the mirrors, it behaves as if it continues in a straight line through the mirror. To simplify the problem, we can use the concept of "virtual points" by reflecting point \( B \) across the x-axis and y-axis.

3. **Reflecting Point \( B \):**
   - Reflect \( B(a, b) \) across the y-axis to get \( B'(-a, b) \).
   - Reflect \( B'(-a, b) \) across the x-axis to get \( B''(-a, -b) \).

4. **Distance Calculation:**
   The total distance the light ray travels is the straight-line distance from \( A(x, y) \) to \( B''(-a, -b) \).

5. **Using the Distance Formula:**
   The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
   \[
   d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]
   Here, \( (x_1, y_1) = (x, y) \) and \( (x_2, y_2) = (-a, -b) \).

6. **Substitute the Coordinates:**
   \[
   d = \sqrt{((-a) - x)^2 + ((-b) - y)^2}
   \]
   Simplify the expression inside the square root:
   \[
   d = \sqrt{(-a - x)^2 + (-b - y)^2}
   \]
   Since squaring a negative number gives a positive result:
   \[
   d = \sqrt{(a + x)^2 + (b + y)^2}
   \]

Conclusion:
The total distance that the light ray travels is:
\[
\boxed{\sqrt{(a + x)^2 + (b + y)^2}}
\]