If the equation of a curve C is transformed to 9 x 2 + 25 y 2 = 225 by the rotation of the coordinate axes…

The original and new coordinates are,

$x=x\cos \frac{\mathrm{\pi }}{4}-y\sin \frac{\mathrm{\pi }}{4}=\frac{x}{\sqrt{2}}-\frac{y}{\sqrt{2}}$

$y=x\sin \frac{\mathrm{\pi }}{4}+y\cos \frac{\mathrm{\pi }}{4}=\frac{x}{\sqrt{2}}+\frac{y}{\sqrt{2}}$

The equation of the curve is,

$9x^2+25y^2=225$

$\Rightarrow {\left(\frac{x-y}{\sqrt{2}}\right)}^2+25{\left(\frac{x+y}{\sqrt{2}}\right)}^2=225$

After simplification,

$9x^2+9y^2+18xy+25x^2+25y^2-50xy=450$

$\Rightarrow 34x^2+34y^2-32xy=450$

$\Rightarrow 17x^2+17y^2-16xy=225$