For an integer $K$,if the point $P(K^2, K+1)$ and the origin $O(0,0)$ lie in the same region between the lines $x+2y-5=0$ and $3x-y+1=0$,then the possible number of such points $P$ is

The equation of the base of an equilateral triangle is $x+y=2$ and one vertex is $(2,-1)$. The length of the side of the triangle is:

If $p$ and $q$ are the lengths of the perpendiculars from the origin to the straight lines $x \sec \alpha + y \operatorname{cosec} \alpha = 10$ and $x \cos \alpha - y \sin \alpha = 10 \cos 2 \alpha$,then $4 p^2 + q^2 =$

Find the distance of the point $(-1, 1)$ from the line $12(x + 6) = 5(y - 2)$. (in $units$)

Let $A(a, b)$,$B(3, 4)$,and $C(-6, -8)$ respectively denote the centroid,circumcentre,and orthocentre of a triangle. Then,the distance of the point $P(2a+3, 7b+5)$ from the line $2x+3y-4=0$ measured parallel to the line $x-2y-1=0$ is

The length of the perpendicular drawn from the origin $(0, 0)$ to the line joining the points $(x', y')$ and $(x'', y'')$ is: