The co-ordinate axes are rotated through an angle $135^{\circ}$. If the co-ordinates of a point $P$ in the…
The co-ordinate axes are rotated through an angle $135^{\circ}$. If the co-ordinates of a point $P$ in the new system are known to be $(4,-3)$, then the co-ordinates of $P$ in the original system are
Let $\left(x_1, y_1\right)$ be the co-ordinate of new system, then
$x_1=x \cos \theta+y \sin \theta$
$y_1=-x \sin \theta+y \cos \theta$
Given that $\left(x_1, y_1\right)=(4,-3)$ and $\theta=135^{\circ}$
$4=x \cos 135^{\circ}+y \sin 135^{\circ}$
$\Rightarrow \quad 4=-\frac{x}{\sqrt{2}}+\frac{y}{\sqrt{2}}$\ldots(\mathrm{i})$
$-3=-x \sin 135^{\circ}+y \cos 135^{\circ}$
$\Rightarrow \quad-3=-\frac{x}{\sqrt{2}}-\frac{y}{\sqrt{2}}$\ldots(\mathrm{ii})$
On adding Eqs. (i) and (ii), we get
$1=-\frac{2 x}{\sqrt{2}}$
$\Rightarrow \quad x=-\frac{1}{\sqrt{2}}$
On subtracting Eqs. (i) and (ii), we get
$7=\frac{2 y}{\sqrt{2}}$
$\Rightarrow \quad y=\frac{7}{\sqrt{2}}$
Thus $(x, y)=\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)$