If the axes are rotated through an angle 45 ° , the coordinates of the point ( 2 2 , - 3 2 ) in the new…

If the coordinate axes are rotated through an angle $\theta$, then a point $P(x, y)$ will have co-ordinate $P(x\text{'}, y\text{'})$

Where $x\text{'}=x\cos \theta +y\sin \theta$ and $y\text{'}=y\cos \theta -x\sin \theta$

Here, $\left(x, y\right)=\left(2\sqrt{2}, -3\sqrt{2}\right)$ and $\theta =45°$

Now, $x\text{'}=2\sqrt{2}\cos 45°-3\sqrt{2}\sin 45°$

$\Rightarrow x\text{'}=2\sqrt{2}\times \frac{1}{\sqrt{2}}-3\sqrt{2}\times \frac{1}{\sqrt{2}}$

$\Rightarrow x\text{'}=2-3=-1$

And, $y\text{'}=-3\sqrt{2}\cos 45°-2\sqrt{2}\sin 45°$

$y\text{'}=-3\sqrt{2}\times \frac{1}{\sqrt{2}}-2\sqrt{2}\times \frac{1}{\sqrt{2}}$

$\Rightarrow y\text{'}=-3-2=-5$

Therefore the coordinates of the given point in the new system is $\left(-1, -5\right)$