একটি সরলরেখা অক্ষদ্বয়ের সাথে $\frac{50}{\sqrt{3}}$ বর্গ একক ক্ষেত্রফল বিশিষ্ট একটি ত্রিভুজ গঠন করে এবং মূলবিন্দু হতে রেখাটির উপর অঙ্কিত লম্ব x-অক্ষের সাথে ${30}^{\circ }$ কোণ উৎপন্ন করে। রেখাটির সমীকরণ কোনটি?

To understand why the chosen answer is correct, we need to analyze the given conditions.

1. **Slope of the Line**: The line forms an angle of ${30}^{°}$ with the x-axis. We know that the slope of a line can be determined by the tangent of the angle it makes with the x-axis. Therefore, the slope (m) can be calculated as:

    
• $m\; =\; tan(30°)\; =\sqrt{3}/3$

2. **Equation of the Line**: The general equation of a line can be written in the slope-intercept form as:

    
• $y\; =\; mx\; +\; c$, where c is the y-intercept.

3. **Using the Point**: The line needs to pass through a point derived from the area condition stated in the question. Given that the area of the triangle formed is defined by the base and height, we can solve for c once we have the slope and the point.

4. **Comparing with Options**: The option a) $\sqrt{3}x+y=10$ can be rearranged to y = $\sqrt{3}x\; +\; 10$, which indicates it has the correct slope based on our earlier calculation and has c = 10, which is reasonable given the area considerations.

Thus, the answer a) is correct because it correctly reflects the slope and is consistent with the conditions of the problem.

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