If the axes are shifted to minus two, minus three and rotated through pi four then transformed equation of 2x 2 + 4 xy -5y 2+ 20x - 22y - 14 = 0 is a X2 - 14 XY + bY 2 = 2. Where a - b =

$\left(\mathrm{h},\mathrm{k}\right)=\left(-2,-3\right),\mathrm{\theta }={45}^0$ using 

$\mathrm{x}-\mathrm{h}=\mathrm{Xcos\theta }-\mathrm{Ysin\theta y}-\mathrm{k}=\mathrm{Xsin\theta }+\mathrm{Ycos\theta }$$$\begin{gathered} \mathrm{Given} \mathrm{equation} \mathrm{is} 2{\mathrm{x}}^2+4\mathrm{xy}-5{\mathrm{y}}^2+20\mathrm{x}-22\mathrm{y}-14=0 \\ new origin =\left(-2,-3\right) \\ Transformed equation after translation is \\ 2{\left(X-2\right)}^2+4\left(X-2\right)\left(Y-3\right)-5{\left(Y-3\right)}^2+20\left(X-2\right)-22\left(Y-3\right)-14=0 \\ 2X^2+4XY-5Y^2=1 \\ axes are rotated throuth {45}^0 \\ x=\frac{X-Y}{\sqrt{2}},y=\frac{X+Y}{\sqrt{2}} \\ Transformed equation after rotation is \\ 2{\left(\frac{X-Y}{\sqrt{2}}\right)}^2+4\left(\frac{X-Y}{\sqrt{2}}\right)\left(\frac{X+Y}{\sqrt{2}}\right)-5{\left(\frac{X+Y}{\sqrt{2}}\right)}^2=1 \\ {X}^2-7Y^2-14XY=2 \\ a=1,b=-7\Rightarrow a-b=8 \end{gathered}$$