If the incentre and the circumcentre of the triangle formed by the lines $x=2$,$4x+3y+7=0$ and $y=3$ are $I$ and $S$ respectively,then $IS=$

If the points $(2k, k), (k, 2k)$ and $(k, k)$ with $k > 0$ enclose a triangle of area $18$ square units,then the centroid of the triangle is equal to

If the mid-points of the sides $BC$,$CA$ and $AB$ of a triangle $ABC$ are respectively $(2,1)$,$(-1,-2)$ and $(3,3)$,then the equation of the side $BC$ is

If the lines $x + 3y = 4$ and $6x - 2y = 7$ are the diagonals of a parallelogram $PQRS$,then $PQRS$ is a:

Suppose the vertices of a triangle are given by $A(0,3)$,$B(-2,0)$,and $C(6,1)$. For the point $(\alpha, \alpha+1)$ to lie inside the triangle,$\alpha$ must lie in the interval:

Let $(\alpha, \beta)$ be the centroid of the triangle formed by the lines $15x - y = 82$,$6x - 5y = -4$,and $9x + 4y = 17$. Then $\alpha + 2\beta$ and $2\alpha - \beta$ are the roots of the equation $...........$.