question_answer

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:

A) 32                                

B)               36    

C)               9                                             

D)               18

Correct Answer: B

Solution :

\[Area=\frac{1}{2}\,\,h.\text{ }k=50\] \[h.\text{ }k=100\] \[h.\text{ }k={{2}^{2}}\,.\,{{5}^{2}}\] Total divisors \[=\left( 2+1 \right)\left( 2+1 \right)=a\] If \[h\,\,>\,\,0,\,\,k>0\] \[But\,\,\,\,\,\,\,\left. \begin{align}   & h>0,\,\,\,\,k<0 \\  & h<0,\,\,\,\,\,k>0 \\  & h<0,\,\,\,\,\,k<0 \\ \end{align} \right\}\]all are possible so that total no. of positive case \[9+9+9+9=36\]