The line joining the points $A(2,0)$ and $B(3,1)$ is rotated through an angle of $45^{\circ}$, about $A$ in…
The line joining the points $A(2,0)$ and $B(3,1)$ is rotated through an angle of $45^{\circ}$, about $A$ in the anti-clockwise direction. The coordinates of $B$ in the new position
$(2, \sqrt{2})$
$(\sqrt{2}, 2)$
$(2,2)$
$(\sqrt{2}, \sqrt{2})$
Solution
Slope of $A B=\frac{1-0}{3-2}=1$
Therefore, $\angle B A X=45^{\circ}$
But
$\angle B A C=45^{\circ}$
$\angle C A X=90^{\circ}$
So, the equation of $A C$ is
$\frac{x-2}{\cos 90^{\circ}}=\frac{y-9}{\sin 90^{\circ}}=r \text { (say) }$
We have $A B=\sqrt{(3-2)^2+(1-0)^2}=\sqrt{2}$
As $A C$ is the new position of $A B$, therefore $A C=A B=\sqrt{2}$ thus, the coordinates of $C$ are given by
$\frac{x-2}{\cos 90^{\circ}}=\frac{y-0}{\sin 90^{\circ}}=\sqrt{2}$
$\Rightarrow \quad x=2, y=\sqrt{2}$
Hence, the coordinates of $C$ are $(2, \sqrt{2})$.
