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 value of

\[ \frac{\left(\frac{1}{p_1}\right)^2 + \left(\frac{1}{p_3}\right)^2}{\left(\frac{1}{p_2}\right)^2} \]

where \( p_1, p_2, p_3 \) are the lengths of the altitudes from the vertices \( A(-1, 1), B(5, 2), C(3, -1) \) of triangle \( ABC \).

### Step 1: Calculate the lengths of the sides of triangle ABC.

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

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

3. **Length of side CA**:
\[ CA = \sqrt{(3 - (-1))^2 + (-1 - 1)^2} = \sqrt{(3 + 1)^2 + (-2)^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \]

### Step 2: Calculate the area of triangle ABC using the determinant formula.

The area \( \Delta \) of triangle ABC can be calculated using the formula:
\[ \Delta = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates:
\[ \Delta = \frac{1}{2} \left| -1(2 - (-1)) + 5((-1) - 1) + 3(1 - 2) \right| \]
\[ = \frac{1}{2} \left| -1(3) + 5(-2) + 3(-1) \right| = \frac{1}{2} \left| -3 - 10 - 3 \right| = \frac{1}{2} \left| -16 \right| = 8 \]

### Step 3: Relate the area to the altitudes.

Using the formula for the area in terms of the base and height:
\[ \Delta = \frac{1}{2} \times \text{base} \times \text{height} \]
Thus, we can express the altitudes as:
\[ p_1 = \frac{2\Delta}{BC}, \quad p_2 = \frac{2\Delta}{CA}, \quad p_3 = \frac{2\Delta}{AB} \]

Calculating the altitudes:
- For \( p_1 \):
\[ p_1 = \frac{2 \times 8}{\sqrt{13}} = \frac{16}{\sqrt{13}} \]

- For \( p_2 \):
\[ p_2 = \frac{2 \times 8}{2\sqrt{5}} = \frac{16}{2\sqrt{5}} = \frac{8}{\sqrt{5}} \]

- For \( p_3 \):
\[ p_3 = \frac{2 \times 8}{\sqrt{37}} = \frac{16}{\sqrt{37}} \]

### Step 4: Substitute the values into the expression.

Now substituting \( p_1, p_2, p_3 \) into the expression:
\[ \frac{\left(\frac{1}{p_1}\right)^2 + \left(\frac{1}{p_3}\right)^2}{\left(\frac{1}{p_2}\right)^2} = \frac{\left(\frac{\sqrt{13}}{16}\right)^2 + \left(\frac{\sqrt{37}}{16}\right)^2}{\left(\frac{\sqrt{5}}{8}\right)^2} \]
Calculating each term:
\[ = \frac{\frac{13}{256} + \frac{37}{256}}{\frac{5}{64}} = \frac{\frac{50}{256}}{\frac{5}{64}} = \frac{50}{256} \times \frac{64}{5} = \frac{50 \times 64}{256 \times 5} = \frac{3200}{1280} = \frac{8}{4} = 2 \]

### Final Answer:
The value of

\[ \frac{\left(\frac{1}{p_1}\right)^2 + \left(\frac{1}{p_3}\right)^2}{\left(\frac{1}{p_2}\right)^2} = 2 \]