If the equation of a curve C is transformed to $9x^2 + 25y^2 = 225$ by the rotation of the coordinate axes about the origin through an angle $\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.$ in the positive direction then the equation of the curve C, before the transformation is

$$\begin{gathered} \mathrm{Given} \mathrm{\theta }=\frac{\mathrm{\pi }}{4} \\ \mathrm{Now} X=\mathrm{x} \cos \frac{\mathrm{\pi }}{4}+\mathrm{y} \sin \frac{\mathrm{\pi }}{4} \\ X=\frac{\mathrm{x}+\mathrm{y}}{\sqrt{2}} \\ \mathrm{and} Y=-\mathrm{x} \sin \frac{\mathrm{\pi }}{4}+\mathrm{y} \cos \frac{\mathrm{\pi }}{4} \\ \Rightarrow Y=\frac{\mathrm{y}-\mathrm{x}}{\sqrt{2}} \end{gathered}$$

Given transformed equation is $9X^2+25Y^2=225$.

Now Original Equation is

$$\begin{gathered} 9\frac{{(\mathrm{x}+\mathrm{y})}^2}{2}+25\frac{{(\mathrm{y}-\mathrm{x})}^2}{2}=225 \\ \Rightarrow 9{\mathrm{x}}^2+9{\mathrm{y}}^2+18\mathrm{xy}+25{\mathrm{y}}^2+25{\mathrm{x}}^2-50\mathrm{xy}=225\times 2 \\ \Rightarrow 34{\mathrm{x}}^2+34{\mathrm{y}}^2-32\mathrm{xy}=225\times 2 \\ \Rightarrow 17{\mathrm{x}}^2-16\mathrm{xy}+17{\mathrm{y}}^2=225 \end{gathered}$$