When the axes are rotated through an angle $45^{\circ}$,the new coordinates of a point $P$ are $(1, -1)$. The coordinates of $P$ in the original system are

The transformed equation of $x^2+y^2=r^2$ when the axes are rotated through an angle $36^{\circ}$ is

The transformed equation of the curve $2x^2+y^2-3x+5y-8=0$ when the origin is translated to the point $(-1, 2)$ is

By rotating the axes about the origin in an anti-clockwise direction by a certain angle,if the equation $x^2+4xy+y^2=1$ is transformed to $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$,then find the value of $\sqrt{\frac{a^2+b^2}{a^2}}$.

Transforming to parallel axes through a point $(p, q)$,the equation $2x^2 + 3xy + 4y^2 + x + 18y + 25 = 0$ becomes $2x^2 + 3xy + 4y^2 = 1$. Then:

$A$ line $L$ makes intercepts $a$ and $b$ on the coordinate axes. The axes are rotated through an angle $\theta$ in the positive direction,keeping the origin fixed. If the line $L$ makes intercepts $p$ and $q$ on the new coordinate axes,then $\frac{1}{a^2}+\frac{1}{b^2}=$