If the circumcentre of the triangle whose vertices are `(0,2),(3,5)and (5,8)`is (h,k) then `4(h^(2)+k^(2))` equals`"_____".`

Answer

To find the value of \( 4(h^2 + k^2) \) where \( (h, k) \) is the circumcenter of the triangle with vertices \( (0, 2) \), \( (3, 5) \), and \( (5, 8) \), we can follow these steps:

### Step 1: Set up the equations based on the circumcenter property
The circumcenter is equidistant from all three vertices of the triangle. Therefore, we can set up the following equations based on the distances from the circumcenter \( (h, k) \) to each vertex:

1. Distance from \( (h, k) \) to \( (0, 2) \):
\[ (h - 0)^2 + (k - 2)^2 = d^2 \]
which simplifies to:
\[ h^2 + (k - 2)^2 = d^2 \tag{1} \]

2. Distance from \( (h, k) \) to \( (3, 5) \):
\[ (h - 3)^2 + (k - 5)^2 = d^2 \]
which simplifies to:
\[ (h - 3)^2 + (k - 5)^2 = d^2 \tag{2} \]

3. Distance from \( (h, k) \) to \( (5, 8) \):
\[ (h - 5)^2 + (k - 8)^2 = d^2 \]
which simplifies to:
\[ (h - 5)^2 + (k - 8)^2 = d^2 \tag{3} \]

### Step 2: Equate the distances
From equations (1) and (2):
\[ h^2 + (k - 2)^2 = (h - 3)^2 + (k - 5)^2 \]

Expanding both sides:
\[ h^2 + (k^2 - 4k + 4) = (h^2 - 6h + 9) + (k^2 - 10k + 25) \]

Simplifying gives:
\[ h^2 + k^2 - 4k + 4 = h^2 - 6h + 9 + k^2 - 10k + 25 \]
\[ -4k + 4 = -6h + 34 - 10k \]
\[ 6h + 6k = 30 \quad \Rightarrow \quad h + k = 5 \tag{4} \]

### Step 3: Use the second pair of distances
Now, equate equations (2) and (3):
\[ (h - 3)^2 + (k - 5)^2 = (h - 5)^2 + (k - 8)^2 \]

Expanding both sides:
\[ (h^2 - 6h + 9) + (k^2 - 10k + 25) = (h^2 - 10h + 25) + (k^2 - 16k + 64) \]

Simplifying gives:
\[ -6h - 10k + 34 = -10h - 16k + 89 \]
\[ 4h + 6k = 55 \quad \Rightarrow \quad 2h + 3k = 27.5 \tag{5} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \( h + k = 5 \) (from equation 4)
2. \( 2h + 3k = 27.5 \) (from equation 5)

From equation (4), we can express \( h \) in terms of \( k \):
\[ h = 5 - k \]

Substituting into equation (5):
\[ 2(5 - k) + 3k = 27.5 \]
\[ 10 - 2k + 3k = 27.5 \]
\[ k = 27.5 - 10 \]
\[ k = 17.5 \]

Substituting \( k \) back into equation (4):
\[ h + 17.5 = 5 \quad \Rightarrow \quad h = 5 - 17.5 = -12.5 \]

### Step 5: Calculate \( 4(h^2 + k^2) \)
Now we have \( h = -12.5 \) and \( k = 17.5 \):
\[ h^2 = (-12.5)^2 = 156.25 \]
\[ k^2 = (17.5)^2 = 306.25 \]
\[ h^2 + k^2 = 156.25 + 306.25 = 462.5 \]
\[ 4(h^2 + k^2) = 4 \times 462.5 = 1850 \]

Thus, the final answer is:
\[ \boxed{1850} \]