Find the transformed equation of $x \cos \theta + y \sin \theta = p$,when the axes are rotated through an angle $\theta$.

$A$ line $L$ has intercepts $a$ and $b$ on the coordinate axes. When the axes are rotated through a given angle $\theta$ keeping the origin fixed,this line $L$ has the intercepts $p$ and $q$. Then

The transformed equation of $3x^2+4xy+y^2-8x-4y-4=0$ is $f(X, Y)=aX^2+2hXY+bY^2+c=0$ when the origin is shifted to a new point by the translation of axes. Then $f(1,1)=$

When the coordinate axes are rotated through an angle $135^{\circ}$,the coordinates of a point $P$ in the new system are known to be $(4, -3)$. Find the coordinates of $P$ in the original system.

If the coordinate axes are rotated through an angle $\frac{\pi}{6}$ about the origin,then the transformed equation of $\sqrt{3} x^2-4 x y+\sqrt{3} y^2=0$ is

By shifting the origin to the point $(-1, 2)$ through translation of axes,if $ax^2+2hxy+by^2+2gx+2fy+c=0$ is the transformed equation of $2x^2-xy+y^2-3x+4y-5=0$,then $2(f+g+h)=$