Suppose the new axes $X, Y$ are generated by rotating the coordinate axes $x, y$ about the origin through an angle of $30^{\circ}$ in the anti-clockwise direction. Then,the transformed equation of $x^2+2 \sqrt{3} xy - y^2 = 2a^2$ with respect to the new axes $X, Y$ is

By rotating the coordinate axes in the positive direction about the origin by an angle $\alpha$,if the point $(1,2)$ is transformed to $\left(\frac{3 \sqrt{3}-1}{2 \sqrt{2}}, \frac{\sqrt{3}+3}{2 \sqrt{2}}\right)$ in the new coordinate system,then $\alpha=$

The point $P(\alpha, \beta)$ with $\alpha > 0, \beta > 0$ undergoes the following transformations successively:
$a)$ Translation by $3$ units in the positive direction of the $x$-axis.
$b)$ Reflection about the line $y = -x$.
$c)$ Rotation of axes through an angle of $\frac{\pi}{4}$ about the origin in the positive direction.
If the final position of the point $P$ is $(-4\sqrt{2}, -2\sqrt{2})$,then find the value of $(\alpha + \beta)$.

The transformed equation of $x^2+6xy+8y^2=10$ when the axes are rotated through an angle $\frac{\pi}{4}$ is:

Find the new coordinates of the reflection of the point $(2, -1)$ about the $y$-axis,when the axes are rotated by $45^{\circ}$ in the negative direction without shifting the origin.

By shifting the origin to the point $(h, 5)$ by the translation of coordinate axes,if the equation $y=x^3-9x^2+cx-d$ transforms to $Y=X^3$,then $\left(d-\frac{c}{h}\right)=$