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 in the xy-plane with one vertex at the origin (0,0) and the other two vertices lying on the coordinate axes with integral coordinates, 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 (0,0), (a,0), and (0,b) is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times b \]
where \( a \) and \( b \) are the lengths along the x-axis and y-axis, respectively.

2. **Setting Up the Equation**:
Given that the area is 50 square units, we can set up the equation:
\[ \frac{1}{2} \times a \times b = 50 \]
Multiplying both sides by 2 gives:
\[ a \times b = 100 \]

3. **Finding Integral Solutions**:
We need to find pairs of positive integers \( (a, b) \) such that their product is 100. To do this, we can find the factor pairs of 100.

4. **Finding Factor Pairs**:
The factor pairs of 100 are:
- \( (1, 100) \)
- \( (2, 50) \)
- \( (4, 25) \)
- \( (5, 20) \)
- \( (10, 10) \)

5. **Considering Negative Values**:
Since \( a \) and \( b \) can also be negative (as they represent coordinates), for each positive pair \( (a, b) \), we can also have:
- \( (-a, b) \)
- \( (a, -b) \)
- \( (-a, -b) \)

6. **Counting the Combinations**:
For each positive pair \( (a, b) \):
- \( (1, 100) \) gives 4 combinations: \( (1, 100), (-1, 100), (1, -100), (-1, -100) \)
- \( (2, 50) \) gives 4 combinations: \( (2, 50), (-2, 50), (2, -50), (-2, -50) \)
- \( (4, 25) \) gives 4 combinations: \( (4, 25), (-4, 25), (4, -25), (-4, -25) \)
- \( (5, 20) \) gives 4 combinations: \( (5, 20), (-5, 20), (5, -20), (-5, -20) \)
- \( (10, 10) \) gives 4 combinations: \( (10, 10), (-10, 10), (10, -10), (-10, -10) \)

7. **Calculating Total Combinations**:
Since the pair \( (10, 10) \) is the same for both coordinates, it contributes only 1 unique triangle configuration, while all other pairs contribute 4. Thus, we calculate:
\[ \text{Total} = 4 + 4 + 4 + 4 + 1 = 17 \]

8. **Final Count**:
Therefore, the total number of triangles in the set \( S \) is:
\[ \text{Total triangles} = 4 \times 4 + 1 = 17 \]

### Final Answer:
The number of elements in the set \( S \) is **20**.