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Question

(D) When the axes are rotated through an angle $	heta = 45^{\circ}$ in the positive direction,the transformation equations are $x = X \cos 	heta - Y

Vedclass · Accepted Answer

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

Vedclass · Answer

(D) When the axes are rotated through an angle $	heta = 45^{\circ}$ in the positive direction,the transformation equations are $x = X \cos 	heta - Y \sin 	heta$ and $y = X \sin 	heta + Y \cos 	heta$.Substituting $	heta = 45^{\circ}$,we get $x = \frac{X-Y}{\sqrt{2}}$ and $y = \frac{X+Y}{\sqrt{2}}$.Given the transformed equation $17x^2 - 16xy + 17y^2 = 225$,we replace $x$ and $y$ with the expressions in terms of $X$ and $Y$:$17\left(\frac{X-Y}{\sqrt{2}}ight)^2 - 16\left(\frac{X-Y}{\sqrt{2}}ight)\left(\frac{X+Y}{\sqrt{2}}ight) + 17\left(\frac{X+Y}{\sqrt{2}}ight)^2 = 225$$\Rightarrow \frac{17}{2}(X^2 + Y^2 - 2XY) - \frac{16}{2}(X^2 - Y^2) + \frac{17}{2}(X^2 + Y^2 + 2XY) = 225$$\Rightarrow \frac{17}{2}X^2 + \frac{17}{2}Y^2 - 17XY - 8X^2 + 8Y^2 + \frac{17}{2}X^2 + \frac{17}{2}Y^2 + 17XY = 225$$\Rightarrow (\frac{17}{2} - 8 + \frac{17}{2})X^2 + (\frac{17}{2} + 8 + \frac{17}{2})Y^2 = 225$$\Rightarrow (17 - 8)X^2 + (17 + 8)Y^2 = 225$$\Rightarrow 9X^2 + 25Y^2 = 225$.Thus,the original equation is $9x^2 + 25y^2 = 225$.

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 $17x^2 - 16xy + 17y^2 = 225$,then the original equation of that curve is:

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