When the coordinate axes are rotated through an angle of $45^{\circ}$ about the origin in the positive…
When the coordinate axes are rotated through an angle of $45^{\circ}$ about the origin in the positive direction, if the transformed equation of a curve is $17 x^2-16 x y+17 y^2=225$, then the original equation of that curve is
$25 x^2+9 y^2=225$
$9 x^2-25 y^2=225$
$25 x^2-16 x y+9 y^2=225$
$9 x^2+25 y^2=225$
Solution
When the axes are rotated through $45^{\circ}$ about the origin in the positive direction then replace $(x, y)$ by $\left(\frac{x}{\sqrt{2}}+\frac{y}{\sqrt{2}}, \frac{-x}{\sqrt{2}}+\frac{y}{\sqrt{2}}\right)$
So, original equation of the curve,
$
\begin{array}{r}
\Rightarrow 17\left(\frac{x}{\sqrt{2}}+\frac{y}{\sqrt{2}}\right)^2-16\left(\frac{x}{\sqrt{2}}+\frac{y}{\sqrt{2}}\right)\left(\frac{-x}{\sqrt{2}}+\frac{y}{\sqrt{2}}\right) \\
+17\left(\frac{-x}{\sqrt{2}}+\frac{y}{\sqrt{2}}\right)^2=225
\end{array}
$
$\begin{aligned} & \Rightarrow \quad 17\left(\frac{x^2}{2}+\frac{y^2}{2}+x y\right)-16\left(\frac{y^2}{2}-\frac{x^2}{2}\right) \\ & +17\left(\frac{x^2}{2}+\frac{y^2}{2}-x y\right)=225 \\ & \Rightarrow \quad 17 \frac{x^2}{2}+17 \frac{y^2}{2}+17 x y-8 y^2+8 x^2 \\ & \quad+17 \frac{x^2}{2}+17 \frac{y^2}{2}-17 x y=225 \\ & \Rightarrow \quad 17 x^2+17 y^2-8 y^2+8 x^2=225 \\ & \Rightarrow \quad 25 x^2+9 y^2=225\end{aligned}$