$A$ line makes intercepts $5$ and $7$ on the coordinate axes. The axes are rotated through an angle $\theta$ in the positive direction about the origin so that the line makes equal intercepts on the new axes,then $|\tan \theta|=$

Statement $(A) :$ The area of the triangle formed by the points $A (20, 22), B (21, 24),$ and $C (22, 23)$ is equal to the area of the triangle formed by the points $P (0, 0), Q (1, 2),$ and $R (2, 1).$
Reason $(R) :$ The area of a triangle remains invariant under the translation of axes.

The coordinate axes are rotated about the origin in the counter-clockwise direction through an angle $60^{\circ}$. If $a$ and $b$ are the intercepts made on the new axes by a straight line whose equation referred to the original axes is $x+y=1$,then $\frac{1}{a^2}+\frac{1}{b^2}=$

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

$A$ 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

If the axes are rotated through an angle $45^{\circ}$,then the coordinates of the point $(4 \sqrt{2}, -6 \sqrt{2})$ in the new system are . . . . . .