Kindly solve this question from coordinate geometry.

Question

The co-ordinates of a point P referred to a rectangular co-ordinate system where O is the origin are $(1,-2)$. The axes are rotated about 0 through angle theta, if coordinates of the new P are $(k-1,k+1)$, then $k^2$?

Answer 1

Rotations preserve length and because it's a rotation with respect to the orgin, the distance from $(1,-2)$ to the orgin is equal to the distance from $(k-1, k+1)$ to the orgin. We can write the equation $$\sqrt{1^2+(-2)^2}=\sqrt{(k-1)^2+(k+1)^2}.$$ Squaring both sides and expanding gives us $$5=2k^2+2.$$ Thus, $$2k^2=3$$ and $$\boxed{k^2=\frac{3}{2}}.$$