When the coordinate axes are rotated by an angle tan - 1 3 4 about the origin, then the equation x 2 + y 2 =…

Let $\theta$ be the angle of rotation, then $\theta ={\tan }^{-1}\left(\frac{3}{4}\right)$.

Now, new coordinate axes are

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

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

Then, 

$X=x\cos \left\{{\tan }^{-1}\left(\frac{3}{4}\right)\right\}-y\sin \left\{{\tan }^{-1}\left(\frac{3}{4}\right)\right\}$

$Y=x\sin \left\{{\tan }^{-1}\left(\frac{3}{4}\right)\right\}+y\cos \left\{{\tan }^{-1}\left(\frac{3}{4}\right)\right\}$

Or

$X=x\left(\frac{4}{5}\right)-y\left(\frac{3}{5}\right)$

$Y=x\left(\frac{3}{5}\right)+y\left(\frac{4}{5}\right)$

On solving, we get

$x=\frac{4X+3Y}{5}$

$y=\frac{4Y-3X}{5}$

Now, putting these values in $x^2+y^2=9$, we get

$\frac{{\left(4X+3Y\right)}^2}{25}+\frac{{\displaystyle {\left(4Y-3X\right)}^2}}{25}=9$

or $X^2+Y^2=9$