Question

The transformed equation of $xy=c^2$ when the axis are rotated through an angle $\frac{\pi }{4}$ (in the anti clockwise direction), is

• A. $X^2-Y^2=c^2$
• B. $X^2-Y^2=2c^2$
• C. $X^2+Y^2=c^2$
• D. $X^2+Y^2=2c^2$

Answer 1

The correct option is B $X^2-Y^2=2c^2$

Original equation is $xy=c^2,\theta =\frac{\pi }{4}$
$x=X\cos \theta -Y\sin \theta =\frac{X-Y}{\sqrt{2}}y=X\sin \theta +Y\cos \theta =\frac{X+Y}{\sqrt{2}}$
Transformed equation is
$\left(\frac{X-Y}{\sqrt{2}}\right)\left(\frac{X+Y}{\sqrt{2}}\right)=c^2\Rightarrow X^2-Y^2=2c^2$