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Question

(C) Let the original coordinates be $(x, y)$ and the new coordinates be $(X, Y)$.The rotation transformation is given by:$x = X \cos 	heta - Y \sin \

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

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(C) Let the original coordinates be $(x, y)$ and the new coordinates be $(X, Y)$.The rotation transformation is given by:$x = X \cos 	heta - Y \sin 	heta$$y = X \sin 	heta + Y \cos 	heta$Given $	heta = \frac{\pi}{4}$,we have $\cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}$.So,$x = \frac{X - Y}{\sqrt{2}}$ and $y = \frac{X + Y}{\sqrt{2}}$.The transformed equation is $9X^2 + 25Y^2 = 225$.Substituting the expressions for $X$ and $Y$ in terms of $x$ and $y$ (or vice versa,here we substitute $X$ and $Y$ back to $x$ and $y$ to find the original equation):$9\left(\frac{x+y}{\sqrt{2}}ight)^2 + 25\left(\frac{-x+y}{\sqrt{2}}ight)^2 = 225$$\frac{9}{2}(x^2 + y^2 + 2xy) + \frac{25}{2}(x^2 + y^2 - 2xy) = 225$$9(x^2 + y^2 + 2xy) + 25(x^2 + y^2 - 2xy) = 450$$34x^2 + 34y^2 - 32xy = 450$Dividing by $2$,we get $17x^2 - 16xy + 17y^2 = 225$.

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 $\frac{\pi}{4}$ in the positive direction,then the equation of the curve $C$ before the transformation is:

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