If `P(1,2), Q(a, b), R(5, 7)` and S (2, 3) are the vertices of a parallelogram, then sum of the squares of the length of its diagonals is `2k^(3)` where k is equal to

Answer

To solve the problem, we need to find the coordinates of point \( Q(a, b) \) such that \( P(1, 2), Q(a, b), R(5, 7), S(2, 3) \) are the vertices of a parallelogram. Then, we will calculate the sum of the squares of the lengths of the diagonals and find the value of \( k \).

### Step 1: Find the coordinates of point \( Q \)

Since \( P \) and \( R \) are opposite vertices of the parallelogram, we can find the coordinates of point \( Q \) using the midpoint formula. The midpoint of diagonal \( PR \) should be equal to the midpoint of diagonal \( QS \).

The midpoint \( M_{PR} \) of \( P(1, 2) \) and \( R(5, 7) \) is given by:
\[ M_{PR} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{1 + 5}{2}, \frac{2 + 7}{2} \right) = \left( 3, 4.5 \right) \]

The midpoint \( M_{QS} \) of \( Q(a, b) \) and \( S(2, 3) \) is given by:
\[ M_{QS} = \left( \frac{a + 2}{2}, \frac{b + 3}{2} \right) \]

Setting \( M_{PR} = M_{QS} \):
\[ \frac{a + 2}{2} = 3 \quad \text{and} \quad \frac{b + 3}{2} = 4.5 \]

From the first equation:
\[ a + 2 = 6 \implies a = 4 \]

From the second equation:
\[ b + 3 = 9 \implies b = 6 \]

Thus, the coordinates of point \( Q \) are \( Q(4, 6) \).

### Step 2: Calculate the lengths of the diagonals \( PR \) and \( QS \)

Using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \):

1. Length of diagonal \( PR \):
\[ PR = \sqrt{(5 - 1)^2 + (7 - 2)^2} = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} \]

2. Length of diagonal \( QS \):
\[ QS = \sqrt{(4 - 2)^2 + (6 - 3)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \]

### Step 3: Calculate the sum of the squares of the lengths of the diagonals

Now we find the sum of the squares of the lengths of the diagonals:
\[ PR^2 + QS^2 = (\sqrt{41})^2 + (\sqrt{13})^2 = 41 + 13 = 54 \]

### Step 4: Relate the sum of squares to \( 2k^3 \)

According to the problem, the sum of the squares of the lengths of the diagonals is given by:
\[ 54 = 2k^3 \]

Dividing both sides by 2:
\[ k^3 = 27 \]

Taking the cube root of both sides:
\[ k = 3 \]

### Final Answer

The value of \( k \) is \( \boxed{3} \).