The line $y = 2x + c$ passes through a point that is equidistant from both the axes and lies in the first quadrant $(x > 0, y > 0)$. Then the value of $c$ is...

The equation of a straight line passing through the points $(-5, -6)$ and $(3, 10)$ is:

$A$ line passing through the point $P(a, 0)$ makes an acute angle $\alpha$ with the positive $x$-axis. Let this line be rotated about the point $P$ through an angle $\frac{\alpha}{2}$ in the clockwise direction. If in the new position,the slope of the line is $2-\sqrt{3}$ and its distance from the origin is $\frac{1}{\sqrt{2}}$,then the value of $3a^2 \tan^2 \alpha - 2\sqrt{3}$ is

The equation of the line passing through the points $(-7, 8)$ and $(5, 2)$ is . . . . . . .

The equations of the sides $AB$,$AC$,and $BC$ of a $\triangle ABC$ are respectively $x-3y=0$,$3x-y=0$,and $x+y+4=0$. If $P$ and $Q$ are points on the line $3x-y+k=0$ passing through $B$ such that $PB:BQ=1:1$,then $k=$

If two vertical poles of heights $20\,m$ and $80\,m$ stand apart on a horizontal plane,then the height (in $m$) of the point of intersection of the lines joining the top of each pole to the foot of the other is: