Transformation of Axes

When the axes are rotated anti-clockwise through an angle $\alpha$, find the transformed equation of $x\cos \alpha +y\sin \alpha =p$.

Find the angle through which the axes are to be rotated so as to remove the $xy$ term in the equation $x^2+4xy+y^2-2x+2y-6=0$.

When the axes are rotated through an angle $60°$, the new coordinates are $\left(2,0\right)$. Find the original coordinates.

When the axes are rotated through an angle $60°$, the new coordinates are $\left(-7,2\right)$. Find the original coordinates.

When the axes are rotated anti clock-wise through an angle $60°$, the new coordinates are $\left(3,4\right)$. Find the original coordinates.

When the axes are rotated anti clock-wise through an angle $30°$, find the new coordinates of the $\left(0,0\right)$.

Find the point to which the origin is to be shifted so as to remove the first degree terms from the equation $4x^2+9y^2-8x+36y+4=0$.

The point to which the origin is shifted and the transformed equation are $(-1,2); x^2+2y^2+16=0$. Find the original equation.

The point to which the origin is shifted and the transformed equation are $(3,-4); x^2+y^2=4$. Find the original equation.

Find the point to which the origin is to be shifted so that the point $\left(3,0\right)$ may change to $\left(2,-3\right)$.

The origin is shifted to $\left(2,3\right)$ by the translation of axes. If the coordinates of a point $P$ change as $\left(0,0\right)$, find the coordinates of $P$ in the original system.

The origin is shifted to $\left(2,3\right)$ by the translation of axes. If the coordinates of a point $P$ change as $\left(-4,3\right)$, find the coordinates of $P$ in the original system.

The origin is shifted to $\left(2,3\right)$ by the translation of axes. If the coordinates of a point $P$ change as $\left(4,5\right)$, find the coordinates of $P$ in the original system.

When the origin is shifted to $(4,-5)$ by the translation of axes, find the coordinates of the $\left(4,-5\right)$ with reference to new axes.

When the origin is shifted to $(4,-5)$ by the translation of axes, find the coordinates of the $\left(-2,4\right)$ with reference to new axes.

When the origin is shifted to $(4,-5)$ by the translation of axes, find the coordinates of the $\left(0,3\right)$ with reference to new axes.

If the coordinates of a point $P$ changes to $\left(2,-6\right)$ when the coordinate axes are rotated through an angle of $135°$, then the coordinates of $P$ in the original system are

The point $P(3,2)$ undergoes the following transformations successively.

(i) Reflection about the line $y=x$.

(ii) Translation to a distance of $3$ units in the positive direction of $x$ -axis.

(iii) Rotation through an angle $\frac{\pi }{4}$ about the origin in the counter-clockwise direction.

Then, the final position of that point is

If co-ordinate axes are so translated such that ordinate of $(4,12)$ becomes zero while abscissa remains same. Then new coordinates of point $(-8,-2)$ are
 

Given the equation $4x^2+2\sqrt{3}xy+2y^2=1.$ Through what angle should the axes be rotated so that the term $xy$ is removed from the transformed equation?