Let S be the set of all triangles in the xy-plane,

Question:

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 $\mathrm{S}$ has area 50 sq. units, then the number of elements in the set $S$ is:

  1. 9

  2. 18

  3. 32

  4. 36


Correct Option: , 4

Solution:

Let $A(\alpha, 0)$ and $B(0, \beta)$

be the vectors of the given triangle $A O B$

$\Rightarrow|\alpha \beta|=100$

$\Rightarrow$ Number of triangles

$=4 \times$ (number of divisors of 100$)$

$=4 \times 9=36$

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