Transformation of Axes

When the coordinate axes are rotated about the origin in the positive direction through an angle $\frac{\pi }{4}$, if the equation $49x^2+25y^2=1225$ is transformed to $p{x}^2+qxy+r{y}^2=t$ and the G.C.D of $p,q,r,t$ is $1$, then

The transformed equation of $3x^2+4xy+y^2-8x-4y-4=0$ is $f\left(X,Y\right)=a{\mathrm{X}}^2+2h\mathrm{XY}+b{\mathrm{Y}}^2+c=0$ when the origin is shifted to a new point by the translation of axes. Then $f\left(1,1\right)=$

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.

When the origin is shifted to $(-1,2)$ by the translation of axes, find the transformed equation of the $x^2+y^2+2x-4y+1=0$, if $X$ and $Y$ are the new coordinates.

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 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
 

If origin is shifted to $\left(h,k\right)$, so that the linear (one degree) terms in the equation $x^2+y^2-4x+6y-7=0$ are eliminated. Then the point $\left(h,k\right)$ is

The new coordinates of a point $\left(4,5\right),$ when the origin is shifted to the point $\left(1,-2\right)$ are

If the equation $x^2+y^2-4x-6y-12=0$ is transformed to $X^2+Y^2=25$ when the axes are translated to a point then the new coordinates of (-3, 5) are

Without changing the direction of coordinate axes, origin is transferred to (h, k), so that the linear (one degree) terms in the equation $x^2+y^2-4x+6y-7=0$ are eliminated. Then the point (h, k) is

The new coordinates of a point $\left(4,5\right)$, when the origin is shifted to the point $\left(1,-2\right)$ are

The point (2, 3) undergoes the following three transformation successively,

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

(ii) Transformation through a distance 2 units along the positive direction of y - axis.

(iii) Rotation through an angle of ${45}^o$ about the origin in the anticlockwise direction.

The final coordinates of points are