When the axes are rotated through an angle 45 ° , the new coordinates of a point P are ( 1 , - 1 ) .…

When axes are rotated through an angle $\theta$, then coordinates are

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

    $=\cos {45}^{\circ }+\sin {45}^{\circ }$

  $=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=\sqrt{2}$

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

  $=\sin {45}^{\circ }-\cos {45}^{\circ }$

    $=\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}=0$

Therefore, the coordinates of $P$ in the original system are $(\sqrt{2},0)$