Question

lf the transformed equation of a curve is $17X^2-16XY+17Y^2=225$ when the axes are rotated through an angle ${45}^o$, then the original equation of the curve is

• A. $25x^2+9y^2=225$
• B. $9x^2+25y^2=225$
• C. $25x^2-9y^2=225$
• D. $9x^2-25y^2=225$

Answer 1

The correct option is A $25x^2+9y^2=225$

Given equation is $17X^2-16XY+17Y^2=225$......eq(i)

We know that

$X=x\cos \theta +y\sin \theta$

$Y=-x\sin \theta +y\cos \theta$

Given $\theta ={45}^0$

$\Rightarrow X=\frac{1}{\sqrt{2}}(x+y)$ 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

$\frac{17}{2}(x+y{)}^2-8(x+y\left)\right(-x+y)+\frac{17}{2}(-x+y{)}^2=225$