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

Options

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

Answer

To solve the problem, we need to find the number of triangles that can be formed with one vertex at the origin (0, 0) and the other two vertices on the x-axis and y-axis, such that the area of each triangle is 50 square units.

### Step-by-Step Solution:

1. **Understanding the Area of the Triangle**:
The area \( A \) of a triangle with vertices at the origin (0, 0), (α, 0), and (0, β) is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times \alpha \times \beta \]
Here, α is the x-coordinate of the vertex on the x-axis, and β is the y-coordinate of the vertex on the y-axis.

2. **Setting Up the Equation**:
We know the area is 50 square units, so we set up the equation:
\[ \frac{1}{2} \times \alpha \times \beta = 50 \]
Multiplying both sides by 2 gives:
\[ \alpha \times \beta = 100 \]

3. **Finding Factor Pairs of 100**:
To find the integer pairs (α, β) such that their product equals 100, we need to find all factor pairs of 100:
- The factor pairs of 100 are:
- (1, 100)
- (2, 50)
- (4, 25)
- (5, 20)
- (10, 10)

4. **Considering Negative Values**:
Since both α and β can be negative (as they are coordinates), we also consider the negative pairs:
- For each positive pair (α, β), we can have:
- (1, 100), (-1, -100), (100, 1), (-100, -1)
- (2, 50), (-2, -50), (50, 2), (-50, -2)
- (4, 25), (-4, -25), (25, 4), (-25, -4)
- (5, 20), (-5, -20), (20, 5), (-20, -5)
- (10, 10), (-10, -10)

5. **Counting the Unique Pairs**:
- For pairs where α and β are different (like (1, 100)), we have 4 combinations (positive and negative).
- For the pair (10, 10), we only have 1 unique triangle since swapping does not create a new triangle.

Therefore, we count:
- For (1, 100): 4 combinations
- For (2, 50): 4 combinations
- For (4, 25): 4 combinations
- For (5, 20): 4 combinations
- For (10, 10): 1 combination

Total combinations:
\[ 4 + 4 + 4 + 4 + 1 = 17 \]

6. **Final Count**:
Each unique pair corresponds to a unique triangle, so the total number of triangles in set \( S \) is:
\[ \text{Number of triangles} = 17 \]

### Conclusion:
The number of elements in the set \( S \) is **17**.