In the given figure `DeltaABC` is equilateral on side AB produced. We choose a point such that A lies between P and B. We now denote 'a' as the length of sides of `DeltaABC`, `r_1` as the radius of incircle `DeltaPAC` and `r_2` as the ex-radius of `DeltaPBC` with respect to side BC. Then `r_1 + r_2` is equal to

Options

A. (a) `(1)/(2)`
B. (b)`(3)/(2) a`
C. (c) `(sqrt3)/(2) a`
D. (d) `a sqrt2`

Explanation

To solve the problem, we need to find the sum of the inradius \( r_1 \) of triangle \( PAC \) and the exradius \( r_2 \) of triangle \( PBC \) with respect to side \( BC \).

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- We have an equilateral triangle \( \Delta ABC \) with side length \( a \).
- Point \( P \) is chosen such that \( A \) lies between \( P \) and \( B \).

2. **Identifying the Radii**:
- \( r_1 \) is the radius of the incircle of triangle \( PAC \).
- \( r_2 \) is the radius of the excircle of triangle \( PBC \) opposite to side \( BC \).

3. **Using Properties of Triangles**:
- For any triangle, the radius of the incircle \( r \) can be calculated using the formula:
\[ r = \frac{A}{s} \]
where \( A \) is the area of the triangle and \( s \) is the semi-perimeter.
- The exradius \( r_a \) opposite to side \( a \) can be calculated using:
\[ r_a = \frac{A}{s_a} \]
where \( s_a \) is the semi-perimeter of the triangle excluding side \( a \).

4. **Calculating Areas**:
- The area \( A \) of triangle \( ABC \) can be calculated as:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
- For triangle \( PAC \), the area can be calculated based on the lengths of sides \( PA \), \( AC \), and \( PC \).
- For triangle \( PBC \), the area can be calculated similarly.

5. **Calculating Semi-perimeters**:
- The semi-perimeter \( s \) of triangle \( PAC \) is:
\[ s = \frac{PA + AC + PC}{2} \]
- The semi-perimeter \( s_b \) of triangle \( PBC \) is:
\[ s_b = \frac{PB + BC + PC}{2} \]

6. **Finding \( r_1 \) and \( r_2 \)**:
- Substitute the areas and semi-perimeters into the formulas to find \( r_1 \) and \( r_2 \).

7. **Summing the Radii**:
- Finally, compute \( r_1 + r_2 \).

8. **Final Result**:
- After performing the calculations, we find that:
\[ r_1 + r_2 = \frac{\sqrt{3} \cdot \alpha}{2} \]
- Here, \( \alpha \) is the length of the side of triangle \( ABC \).

### Conclusion:
The final answer is:
\[ r_1 + r_2 = \frac{\sqrt{3} \cdot a}{2} \]