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 given problem step by step, we will analyze the equations and use the provided condition to find the value of 't'.

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

The system has a unique solution, which implies that the determinant of the coefficients of the first two equations must be non-zero.

### Step 2: Find the determinant condition for unique solutions
The determinant \( D \) of the coefficients from the first two equations can be calculated as follows:
\[ D = \begin{vmatrix} a & h \\ h & b \end{vmatrix} = ab - h^2 \]
For the system to have a unique solution, we require:
\[ D \neq 0 \implies ab - h^2 \neq 0 \]

### Step 3: Use the given condition
We are given the condition:
\[ \frac{abc + 2fgh - af^2 - bg^2 - ch^2}{h^2 - ab} = 8 \]
We can rearrange this to find \( t \):
\[ abc + 2fgh - af^2 - bg^2 - ch^2 = 8(h^2 - ab) \]

### Step 4: Express \( t \) in terms of the other variables
From Equation 3, we can express \( t \):
\[ t = - (ax^2 + 2hxy + by^2 + 2gx + 2fy + c) \]
Substituting the expression we derived from the condition into this equation gives us:
\[ t = - (abc + 2fgh - af^2 - bg^2 - ch^2) + 8(h^2 - ab) \]

### Step 5: Substitute and simplify
Now, we can substitute the expression from the condition into our expression for \( t \):
\[ t = - (abc + 2fgh - af^2 - bg^2 - ch^2) + 8(h^2 - ab) \]
This means that \( t \) can be expressed as:
\[ t = - (abc + 2fgh - af^2 - bg^2 - ch^2) + 8h^2 - 8ab \]

### Step 6: Final value of \( t \)
Since we know that the left-hand side of the condition equals \( 8(h^2 - ab) \), we can conclude that:
\[ t = 8(h^2 - ab) - (abc + 2fgh - af^2 - bg^2 - ch^2) \]
Thus, substituting the known values will yield the final value of \( t \).

### Conclusion
The value of \( t \) is determined by the relationship established through the equations and the condition provided.