Point $(-1,2)$ is changed to $(\mathrm{a}, \mathrm{b})$ when the origin is shifted to the point $(2,-1)$ by…
Point $(-1,2)$ is changed to $(\mathrm{a}, \mathrm{b})$ when the origin is shifted to the point $(2,-1)$ by translation of axes, Point $(a, b)$ is changed to $(c, d)$ when the axes are rotated through an angle of $45^{\circ}$ about the new origin. $(c, d)$ is changed to ( $e$, $f)$ when $(c, d)$ is reflected through $y=x$. Then $(e, f)=$
$(-3,3)$
$(0,3 \sqrt{2})$
$(3 \sqrt{2}, 0)$
$(1,2)$
Solution
Since, after translation
So, $a=-1-2=-3, b=2-(-1)=3 \Rightarrow(a, b)=(-3,3)$
Since, $\theta=45^{\circ}$
$\Rightarrow c=-3 \cos \left(45^{\circ}\right)+3 \sin \left(45^{\circ}\right)=\frac{-3}{\sqrt{2}}+\frac{3}{\sqrt{2}}=0$
and $d=-(-3) \sin \left(45^{\circ}\right)+3 \cos \left(45^{\circ}\right)=\frac{3}{\sqrt{2}}+\frac{3}{\sqrt{2}}=3 \sqrt{2}$
So, $(c, d)=(0,3 \sqrt{2})$
Now, reflection of point $(0,3 \sqrt{2})$ about $y=x$ is $(e, f)=(3 \sqrt{2}, 0)$.