When the axes are rotated through an angle $45^{\circ},$ the transformed equation of a curve is $17 x^{2} - 16 x y + 17 y^{2} = 225.$ Find the original equation of the curve.

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Solution

Given equation is $17 X^{2} - 16 X Y + 17 Y^{2} = 225$ ........eq(i)
We know that
$X = x c o s θ + y s i n θ$
$Y = - x s i n θ + y c o s θ$
Given $θ = 45^{\circ}$
$\Rightarrow X = \frac{1}{\sqrt{2}} (x + y)$ and $Y = \frac{1}{\sqrt{2}} (- x + y)$
substitute the values X and Y in the given eq(i) we get the following results
So, the transformed equation is
$\frac{17}{2} (x + y)^{2} - 8 (x + y) (- x + y) + \frac{17}{2} (- x + y)^{2} = 225$

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When the axes are rotated through an angle 45∘, the transformed equation of a curve is 17x2−16xy+17y2=225. Find the original equation of the curve.

Given equation is 17X2−16XY+17Y2=225 ........eq(i)We know thatX=xcosθ+ysinθY=−xsinθ+ycosθGiven θ=45∘⇒X=1√2(x+y) and Y=1√2(−x+y)substitute the values X and Y in the given eq(i) we get the following resultsSo, the transformed equation is172(x+y)2−8(x+y)(−x+y)+172(−x+y)2=225

When the axes are rotated through an angle $45^{\circ},$ the transformed equation of a curve is $17 x^{2} - 16 x y + 17 y^{2} = 225.$ Find the original equation of the curve.