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

$A$ straight line which makes equal intercepts on positive $X$ and $Y$ axes and which is at a distance $1$ unit from the origin intersects the straight line $y=2x+3+\sqrt{2}$ at $(x_0, y_0)$. Then $2x_0+y_0$ is equal to

The number of straight lines that can be drawn through the point $(2, 5)$ which form a triangle of area $24 \text{ sq. units}$ with the coordinate axes is:

For $\lambda, \mu \in R$,$(x-2y-1)+\lambda(3x+2y-11)=0$ and $(3x+4y-11)+\mu(-x+2y-3)=0$ represent two families of lines. If the equation of the line common to both the families is $ax+by-5=0$,then $2a+b=$

If a line passing through the point $P(1, 1)$ makes an angle $\theta$ with the positive direction of the $x$-axis and intersects the coordinate axes at $A$ and $B$ such that $AB = OP$,where $O$ is the origin,then what is the value of $\tan\theta + \cot\theta$?

$A$ line $4x + y = 1$ passes through the point $A(2, -7)$ and meets the line $BC$,whose equation is $3x - 4y + 1 = 0$,at the point $B$. The equation of the line $AC$ such that $AB = AC$ is