Let the lengths of the altitudes from the vertices `A(-1, 1), B(5, 2), C(3, -1)` of `DeltaABC` are `p_(1), p_(2), p_(3)` units respectively then the value of `(((1)/(p_(1)))^(2)+((1)/(p_(3)))^(2))/(((1)/(p_(2)))^(2))` is equal to

Answer

To solve the problem, we need to find the lengths of the altitudes from the vertices \( A(-1, 1) \), \( B(5, 2) \), and \( C(3, -1) \) of triangle \( \Delta ABC \) and then compute the expression given in the question.

### Step 1: Calculate the lengths of the sides of the triangle \( ABC \)

1. **Length of side \( AB \)**:
\[ AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(5 - (-1))^2 + (2 - 1)^2} = \sqrt{(5 + 1)^2 + (2 - 1)^2} = \sqrt{6^2 + 1^2} = \sqrt{36 + 1} = \sqrt{37} \]

2. **Length of side \( BC \)**:
\[ BC = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2} = \sqrt{(3 - 5)^2 + (-1 - 2)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \]

3. **Length of side \( CA \)**:
\[ CA = \sqrt{(x_A - x_C)^2 + (y_A - y_C)^2} = \sqrt{(-1 - 3)^2 + (1 - (-1))^2} = \sqrt{(-4)^2 + (2)^2} = \sqrt{16 + 4} = \sqrt{20} \]

### Step 2: Calculate the area of triangle \( ABC \)

Using the determinant formula for the area of a triangle given by vertices \( (x_1, y_1) \), \( (x_2, y_2) \), \( (x_3, y_3) \):
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates:
\[ \text{Area} = \frac{1}{2} \left| -1(2 - (-1)) + 5((-1) - 1) + 3(1 - 2) \right| \]
\[ = \frac{1}{2} \left| -1(3) + 5(-2) + 3(-1) \right| = \frac{1}{2} \left| -3 - 10 - 3 \right| = \frac{1}{2} \left| -16 \right| = 8 \]

### Step 3: Calculate the altitudes \( p_1, p_2, p_3 \)

1. **Altitude from \( A \) (to side \( BC \))**:
\[ p_1 = \frac{2 \times \text{Area}}{BC} = \frac{2 \times 8}{\sqrt{13}} = \frac{16}{\sqrt{13}} \]

2. **Altitude from \( B \) (to side \( CA \))**:
\[ p_2 = \frac{2 \times \text{Area}}{CA} = \frac{2 \times 8}{\sqrt{20}} = \frac{16}{\sqrt{20}} = \frac{16}{2\sqrt{5}} = \frac{8}{\sqrt{5}} \]

3. **Altitude from \( C \) (to side \( AB \))**:
\[ p_3 = \frac{2 \times \text{Area}}{AB} = \frac{2 \times 8}{\sqrt{37}} = \frac{16}{\sqrt{37}} \]

### Step 4: Calculate the expression

We need to find:
\[ \frac{\left( \frac{1}{p_1} \right)^2 + \left( \frac{1}{p_3} \right)^2}{\left( \frac{1}{p_2} \right)^2} \]

Calculating each term:
1. **For \( p_1 \)**:
\[ \frac{1}{p_1} = \frac{\sqrt{13}}{16} \implies \left( \frac{1}{p_1} \right)^2 = \frac{13}{256} \]

2. **For \( p_2 \)**:
\[ \frac{1}{p_2} = \frac{\sqrt{5}}{8} \implies \left( \frac{1}{p_2} \right)^2 = \frac{5}{64} \]

3. **For \( p_3 \)**:
\[ \frac{1}{p_3} = \frac{\sqrt{37}}{16} \implies \left( \frac{1}{p_3} \right)^2 = \frac{37}{256} \]

Now substituting back into the expression:
\[ \frac{\frac{13}{256} + \frac{37}{256}}{\frac{5}{64}} = \frac{\frac{50}{256}}{\frac{5}{64}} = \frac{50}{256} \times \frac{64}{5} = \frac{50 \times 64}{256 \times 5} = \frac{3200}{1280} = 2.5 \]

### Final Answer
The value of the expression is \( 2.5 \).