`P_(1), P_(2), P_(3)` are altitudes of a triangle ABC from the vertices A, B, C and `Delta` is the area of the triangle,
The value of `P_(1)^(-1) + P_(2)^(-1) + P_(3)^(-1)` is equal to-

Options

A. `(s-a)/Δ
B. `(s-b)/Δ
C. `(s-c)/Δ
D. `s/Δ

Answer

To solve the problem, we need to find the value of \( P_1^{-1} + P_2^{-1} + P_3^{-1} \), where \( P_1, P_2, P_3 \) are the altitudes of triangle \( ABC \) from vertices \( A, B, C \) respectively, and \( \Delta \) is the area of the triangle.

### Step-by-Step Solution:

1. **Understand the relationship between area and altitudes**:
The area \( \Delta \) of triangle \( ABC \) can be expressed in terms of its base and height (altitude). Thus, we have:
\[ \Delta = \frac{1}{2} \times A \times P_1 \quad (1) \]
\[ \Delta = \frac{1}{2} \times B \times P_2 \quad (2) \]
\[ \Delta = \frac{1}{2} \times C \times P_3 \quad (3) \]

2. **Rearranging the equations**:
From equation (1), we can express \( P_1 \):
\[ P_1 = \frac{2\Delta}{A} \quad (4) \]
From equation (2), we can express \( P_2 \):
\[ P_2 = \frac{2\Delta}{B} \quad (5) \]
From equation (3), we can express \( P_3 \):
\[ P_3 = \frac{2\Delta}{C} \quad (6) \]

3. **Finding the reciprocals of the altitudes**:
Now, we find the reciprocals:
\[ P_1^{-1} = \frac{A}{2\Delta} \quad (7) \]
\[ P_2^{-1} = \frac{B}{2\Delta} \quad (8) \]
\[ P_3^{-1} = \frac{C}{2\Delta} \quad (9) \]

4. **Adding the reciprocals**:
Now, we can add the reciprocals of the altitudes:
\[ P_1^{-1} + P_2^{-1} + P_3^{-1} = \frac{A}{2\Delta} + \frac{B}{2\Delta} + \frac{C}{2\Delta} \]
This simplifies to:
\[ = \frac{A + B + C}{2\Delta} \quad (10) \]

5. **Using the semi-perimeter**:
The semi-perimeter \( S \) of triangle \( ABC \) is given by:
\[ S = \frac{A + B + C}{2} \]
Therefore, we can rewrite equation (10) as:
\[ P_1^{-1} + P_2^{-1} + P_3^{-1} = \frac{2S}{2\Delta} = \frac{S}{\Delta} \quad (11) \]

6. **Final expression**:
Thus, the final value of \( P_1^{-1} + P_2^{-1} + P_3^{-1} \) is:
\[ P_1^{-1} + P_2^{-1} + P_3^{-1} = \frac{S}{\Delta} \]

### Conclusion:
The value of \( P_1^{-1} + P_2^{-1} + P_3^{-1} \) is \( \frac{S}{\Delta} \).