Point $(-1, 2)$ is changed to $(a, b)$ 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. Point $(c, d)$ is changed to $(e, f)$ when $(c, d)$ is reflected through the line $y = x$. Then $(e, f) =$

When the coordinate axes are rotated about the origin in the positive direction through an angle $\frac{\pi}{4}$,if the equation $25x^2+9y^2=225$ is transformed to $\alpha x^2+\beta xy+\gamma y^2=\delta$,then $(\alpha+\beta+\gamma-\sqrt{\delta})^2=$

By shifting the origin to the point $(2, 3)$ and then rotating the coordinate axes through an angle $\theta$ in the counter-clockwise direction,if the equation $3x^2 + 2xy + 3y^2 - 18x - 22y + 50 = 0$ is transformed to $4X^2 + 2Y^2 - 1 = 0$,then the angle $\theta =$

Find the coordinates of $M$ in the original system if the point $M$ changes to $(4, -3)$ when the axes are rotated through an angle of $135^{\circ}$.

The origin is translated to $(1,2)$. The point $(7,5)$ in the old system undergoes the following transformations successively.
$I$. Moves to the new point under the given translation of origin.
$II$. Translated through $2$ units along the negative direction of the new $X$-axis.
$III$. Rotated through an angle $\frac{\pi}{4}$ about the origin of the new system in the clockwise direction. The final position of the point $(7,5)$ is

If the axes are rotated through an angle $45^{\circ}$ about the origin in an anticlockwise direction,then the transformed equation of $y^2=4ax$ is