The coordinate axes are rotated through an angle $22°$ about the origin. If the equation $4{\mathrm{x}}^2+12\mathrm{xy}+9{\mathrm{y}}^2+6\mathrm{x}+9\mathrm{y}+2=0$ changes to ${\mathrm{ax}}^2+2\mathrm{hxy}+{\mathrm{by}}^2+2\mathrm{gx}+2\mathrm{fy}+\mathrm{c}=0$ then value of $\frac{{\mathrm{g}}^2-\mathrm{ac}}{{\mathrm{a}}^2+{\mathrm{h}}^2}=$

$$\begin{gathered} Given \theta ={22}^0 and the equation 4x^2+12xy+9y^2+6x+9y+2=0\to \left(1\right) \\ changes as a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0 \\ where a=4,2h=12,b=9,2g=6,2f=9,c=2 \\ Equation \left(1\right) represents a pair of parallel lines \\ In rotation of any parallel lines are not change \\ Now \frac{g^2-ac}{a^2+h^2}=\frac{9-8}{16+36}=\frac{1}{52} \end{gathered}$$