The number of triangles formed by the lines $x-y+3=0$,$2x-y+3=0$,$3x-y+2=0$,and $x+y-3=0$ is:

If the line $3x + 4y - 24 = 0$ intersects the $x-$axis at the point $A$ and the $y-$axis at the point $B$,then the incentre of the triangle $OAB$,where $O$ is the origin,is

Consider the set of all triangles $OPQ$ where $O$ is the origin and $P, Q$ are distinct points in the plane with non-negative integral coordinates $(x, y)$ such that $5x + y = 99$. The number of such distinct triangles whose area is a positive integer is:

Suppose $ABOC$ is a rhombus in the first quadrant with $O$ being the origin. If the vertices $B$ and $C$ of $\triangle ABC$ lie respectively on $y=\frac{4}{3}x$ and $y=0$,and the side $BC$ passes through $\left(\frac{2}{3}, \frac{2}{3}\right)$,then the mid-point of $BC$ is

Let $A(1, 1)$,$B(-4, 3)$,and $C(-2, -5)$ be the vertices of a triangle $ABC$. Let $P$ be a point on the side $BC$,and let $\Delta_{1}$ and $\Delta_{2}$ be the areas of triangle $APB$ and triangle $ABC$,respectively. If $\Delta_{1} : \Delta_{2} = 4 : 7$,then find the area enclosed by the lines $AP$,$AC$,and the $x$-axis.

If the sum of the distances of a point from two perpendicular lines in a plane is $1$,then its locus is