# Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in S has area 50 sq. units, then the number of elements in the set S is:-

Let the coordinates of the triangles be $(0,0),(a,0),(0,b)$ where $a,b \in I$

Area of the triangle = $\frac{1}{2}|ab| = 50$

$\Rightarrow |ab| = 100$

$100$ as a product of two integers in first quadrant only

$n\{(1,100),(2,50),(4,25), (5,20),(10,10),(20,5),(25,4),(50,2),(100,1)\}$

= 9

$\therefore$ In all four quadrants = $9\times4 =36$

Therefore, there are $36$ elements in the set S

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