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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 wi

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:-

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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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