Question

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

Answer 1

Area $=\frac{1}{2}\begin{vmatrix}0 & 0 & 1 \\ x_1 & 99-x_1 & 1 \\ x_2 & 99-x_2 & 1\end{vmatrix}$
$ =\frac{1}{2}\left[x_1\left(99-x_2\right)-x_2\left(99-x_1\right)\right]$
$\text { Area }= \frac{1}{2}\left|\left[\left(x_1-x_2\right) 99\right]\right|$
Area is an integer then both $x_1$ and $x_2$ are simultaneously either even or both odd.

hence ${ }^{10} C _2+{ }^{10} C _2=2 \cdot{ }^{10} C _2=90$