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

• A. $9$
• B. $18$
• C. $32$
• D. $36$

Answer 1

The correct option is D $36$

Let $A(\alpha ,0)$ and $B(0,\beta )$
be the vectors of the given triangle $AOB$

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

$\Rightarrow$ Number of triangle

$=4\times$ (number of divisors of $100$) (because there will be 4 triangles in 4 quadrants)

$4\times 9=36$