The equation of the straight line which is perpendicular to the line $5x - 2y = 7$ and passing through the point of intersection of the lines $2x + 3y - 1 = 0$ and $3x + 4y - 6 = 0$ is

The equation of the straight line in the normal form which is parallel to the lines $x+2y+3=0$ and $x+2y+8=0$ and divides the distance between these two lines in the ratio $1:2$ internally is

If $p$ and $q$ are the lengths of the perpendiculars from the origin on the lines,$x \operatorname{cosec} \alpha - y \sec \alpha = k \cot 2 \alpha$ and $x \sin \alpha + y \cos \alpha = k \sin 2 \alpha$ respectively,then $k^{2}$ is equal to :

Let the line $2x + 3y = 18$ intersect the $Y$-axis at $B$. Suppose $C(\neq B)$,with coordinates $(a, b)$,is a point on the line such that $PB = PC$,where $P = (10, 10)$. Then,$8a + 2b$ equals

$A$ straight line $L$ passes through the point of intersection of the lines $x - y + 1 = 0$ and $2x + y - 7 = 0$. If $L$ intersects the positive $x$-axis at $A(a, 0)$ and the positive $y$-axis at $B(0, b)$, then the minimum area of the triangle $OAB$ (where $O$ is the origin) is ....

The point $(4,1)$ undergoes the following transformations successively:
$(i)$ Reflection in the line $x-y=0$
(ii) Shifting through a distance of $2$ units along the positive $X$-axis
(iii) Projection on the $X$-axis
The coordinates of the point in its final position are