If the system of equations ax + hy + g = 0, hx + by + f = 0 and ${\mathrm{ax}}^2+2\mathrm{hxy}+{\mathrm{by}}^2+2\mathrm{gx}+2\mathrm{fy}+\mathrm{c}+\mathrm{t}=0$has a unique solution, then

Clearly $\mathrm{ab}-{\mathrm{h}}^2\ne 0$$[\therefore$first two equations are solvable for x, y]
Third equation can be written as
x (ax + hy + g) + y (hx + by + f) + gx + fy + c + t = 0
Since any solution of first two must be sol. of the third equation.
$\therefore$ given system of equations can be written as
ax + hy + g = 0
hx + by + f = 0        gx + fy + c + t = 0
$\begin{array}{l}\therefore \left|\begin{array}{lll}\mathrm{a}& \mathrm{h}& \mathrm{g}\\ \mathrm{h}& \mathrm{b}& \mathrm{f}\\ \mathrm{g}& \mathrm{f}& \mathrm{c}+\mathrm{t}\end{array}\right|=0\Rightarrow \left|\begin{array}{lll}\mathrm{a}& \mathrm{h}& \mathrm{g}\\ \mathrm{h}& \mathrm{b}& \mathrm{f}\\ \mathrm{g}& \mathrm{f}& \mathrm{c}\end{array}\right|+\left|\begin{array}{lll}\mathrm{a}& \mathrm{h}& 0\\ \mathrm{h}& \mathrm{b}& 0\\ \mathrm{g}& \mathrm{f}& \mathrm{t}\end{array}\right|=0\\ \Rightarrow \left(\mathrm{abc}+2\mathrm{fgh}-{\mathrm{af}}^2-{\mathrm{bg}}^2-{\mathrm{ch}}^2\right)+\mathrm{t}\left(\mathrm{ab}-{\mathrm{h}}^2\right)=0\\ \Rightarrow \mathrm{t}=\frac{\mathrm{abc}+2\mathrm{fgh}-{\mathrm{af}}^2-{\mathrm{bg}}^2-{\mathrm{ch}}^2}{\left({\mathrm{h}}^2-\mathrm{ab}\right)}\end{array}$