The system of equations $x+2y=3$ and $3x+6y=a-2$ has no solution if:

The number of points on the line $x+y=4$ which are at a unit distance from the line $2x+2y=5$ is

What is the perpendicular distance of the line $3x + 4y = 10$ from the origin?

An equilateral triangle is constructed between the lines $\sqrt{3} x+y-6=0$ and $\sqrt{3} x+y+9=0$ with the base on one line and the vertex on the other. The area (in sq. units) of the triangle so formed is

The number of straight lines that can be drawn through the point $(-3, 4)$ which are at a distance of $5$ units from the point $(2, -8)$ is

If $p_{1}$ and $p_{2}$ are the lengths of perpendiculars from the origin to the lines $x \sin \theta + y \cos \theta = 5 \cos 2 \theta$ and $x \operatorname{cosec} \theta + y \sec \theta - 5 = 0$ respectively,then $p_{1}^{2} + 4 p_{2}^{2} = $