Determine if the points $(1,5), (2,3),$ and $(-2,-11)$ are collinear.

In what ratio does the point $(-4, 6)$ divide the line segment joining the points $A(-6, 10)$ and $B(3, -8)$?

If $(1, 2), (4, y), (x, 6)$ and $(3, 5)$ are the vertices of a parallelogram taken in order,find $x$ and $y$.

The Class $X$ students of a secondary school in Krishinagar have been allotted a rectangular plot of land for their gardening activity. Saplings of Gulmohar are planted on the boundary at a distance of $1\, m$ from each other. There is a triangular grassy lawn in the plot as shown in the Figure. The students are to sow seeds of flowering plants on the remaining area of the plot.
What will be the coordinates of the vertices of $\Delta PQR$ if $C$ is the origin?
Also,calculate the area of the triangle in this case.

Find the coordinates of a point $A,$ where $AB$ is the diameter of a circle whose centre is $(2,-3)$ and $B$ is $(1,4).$

Show that the points $(1,7), (4,2), (-1,-1)$ and $(-4,4)$ are the vertices of a square.