If the transformed equation of a curve is $17x^2-16xy+17y^2=225$  when axes are rotated through an angle ${45}^0,$ then the original system of the curve is

$X=x\cos {45}^0+y\sin {45}^0=\frac{x+y}{\sqrt{2}},Y=-x\sin 45+y\cos {45}^0=\frac{y-x}{\sqrt{2}}$

The original equation of the curve is $17{\left(\frac{x+y}{\sqrt{2}}\right)}^2-16\left(\frac{x+y}{\sqrt{2}}\right)\left(\frac{y-x}{\sqrt{2}}\right)+17{\left(\frac{y-x}{\sqrt{2}}\right)}^2=225$
$\Rightarrow \frac{17}{2}\left[{\left(x+y\right)}^2+{\left(y-x\right)}^2\right]-\frac{16}{2}\left(y^2-x^2\right)=225\Rightarrow 25x^2+9y^2=225$