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Let S be the set of all triangles in the...

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 eq. units, then the number of elements in the set S is

A

32

B

9

C

18

D

36

Text Solution

Verified by Experts

The correct Answer is:
D
`|(1)/(2)xy|=50`
`|x||y|=100`
Let's find the total number of solutions of
`xy=100` under the assumption x > 0 and y > 0
`xy=2^(2).5^(2)`
`2=2^(a_(1))5^(a_(2))`
`=2^(a_(1))5^(a_(2)) and y=2^(b_(1))5^(b_(2))`
`x.y=2^(a_(1)+b_(2)) 5(b_(1)+b_(2))=2^(2).5^(2)`
`a_(1)+b_(1)=2 and a_(2)+a_(1)+b_(1)=2`
Total numbers of solution `=(.^(2+2-1)C_(2-1).^(2+2-1)C_(2-1))=9`
But for every (x, y), there is `(x, -y), (-x, y)" as "(-x, -y).`
Therefore, total number of solutions = 36.
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