If the normal drawn from the origin to the straight line $2x + 7y + 6 = 0$ makes an angle $\theta$ with the positive $X$-axis,then $\theta =$

The distance from the origin to the image of $(1,1)$ with respect to the line $x+y+5=0$ is (in $\sqrt{2}$)

Let $L$ be the line $y = 2x$ in the two-dimensional plane.
Statement $1$: The image of the point $(0, 1)$ in $L$ is the point $\left( \frac{4}{5}, \frac{3}{5} \right)$.
Statement $2$: The points $(0, 1)$ and $\left( \frac{4}{5}, \frac{3}{5} \right)$ lie on opposite sides of the line $L$ and are at equal distance from it.

The coordinates of the foot of the perpendicular drawn from the point $(3,4)$ on the line $2x+y-7=0$ are

The foot of the perpendicular drawn from $(2, 4)$ to the line $x + y = 1$ is

Let the mirror image of point $A(1, 4)$ in the line $y = x$ be point $B$; the mirror image of point $B$ in the line $y = -x$ be $C$; and the mirror image of $C$ in the $x$-axis be $D$. Then,the area of triangle $ABD$ is ............... $sq. \, units$.