Permutations and Combinations 4 Question 2
2. Let $S$ be the set of all triangles in the $x y$-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
(2019 Main, 9 Jan II)
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Answer:
Correct Answer: 2. (a)
Solution:
- According to given information, we have the following figure.
(Note that as $a$ and $b$ are integers so they can be negative also). Here $O(0,0), A(a, 0)$ and $B(0, b)$
are the three vertices of the triangle.
Clearly, $O A=|a|$ and $O B=|b|$.
$\therefore$ Area of $\triangle O A B=\frac{1}{2}|a||b|$.
But area of such triangles is given as 50 sq units.
$$ \begin{aligned} & \therefore \quad \frac{1}{2}|a||b|=50 \\ & \Rightarrow \quad|a||b|=100=2^{2} \cdot 5^{2} \end{aligned} $$
Number of ways of distributing two 2’s in $|a|$ and $|b|=3$
| $|a|$ | $|b|$ |
|---|---|
| 0 | 2 |
| 1 | 1 |
| 2 | 0 |
$\Rightarrow 3$ ways
Similarly, number of ways of distributing two 5’s in $|a|$ and $|b|=3$ ways.
$\therefore$ Total number of ways of distributing 2’s and 5’s $=3 \times 3=9$ ways
Note that for one value of $|a|$, there are 2 possible values of $a$ and for one value of $|b|$, there are 2 possible values of $b$.
$\therefore$ Number of such triangles possible $=2 \times 2 \times 9=36$.
So, number of elements in $S$ is 36 .
