If the system of equations
ax+hy+g=0 ….(i)
hx+by+f=0…(ii)
and `ax^2+2hxy+by^2 + 2gx+ 2fy+c+t=0`….(iii)
has a unique solution and `(abc+2fgh-af^2-bg^2-ch^2)/(h^2-ab)=8` , find the value of 't'.

Answer

To solve the problem step by step, we start with the given equations and conditions.

### Step 1: Write down the equations
We have the following system of equations:
1. \( ax + hy + g = 0 \) (Equation i)
2. \( hx + by + f = 0 \) (Equation ii)
3. \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c + t = 0 \) (Equation iii)

### Step 2: Rewrite the third equation
We can rewrite the third equation by grouping terms:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c + t = 0 \]
This can be rearranged as:
\[ ax^2 + 2hxy + by^2 + (2g)x + (2f)y + (c + t) = 0 \]

### Step 3: Identify the unique solution condition
For the system of equations to have a unique solution, the determinant of the coefficients of the first two equations must be non-zero. The determinant is given by:
\[ D = \begin{vmatrix} a & h \\ h & b \end{vmatrix} = ab - h^2 \]
For a unique solution, we need \( D \neq 0 \), which implies \( ab - h^2 \neq 0 \).

### Step 4: Use the given condition
We are given the condition:
\[ \frac{abc + 2fgh - af^2 - bg^2 - ch^2}{h^2 - ab} = 8 \]
From this, we can express the numerator:
\[ abc + 2fgh - af^2 - bg^2 - ch^2 = 8(h^2 - ab) \]

### Step 5: Rearranging the equation
We can rearrange the equation to isolate \( t \):
\[ abc + 2fgh - af^2 - bg^2 - ch^2 = 8h^2 - 8ab \]
This implies:
\[ t = 8h^2 - 8ab - (abc + 2fgh - af^2 - bg^2 - ch^2) \]

### Step 6: Substitute the values
To find \( t \), we can substitute the values we have from the condition:
\[ t = 8h^2 - 8ab - (abc + 2fgh - af^2 - bg^2 - ch^2) \]
Given that the left-hand side equals 8, we can simplify:
\[ t = 8 \]

### Conclusion
Thus, the value of \( t \) is:
\[ \boxed{8} \]