A balloon is slanting down in a straight line passes vertically above two points A and B on a horizontal plane 1000 meters apart. When above A it has an altitude `60^(@)` as seen from B and when above B, altitude is `30^(@)` as seen from A. What is the distance of the point from point A where the balloon strike the plane.

Options

A. 500 m
B. `1000sqrt(3)` m
C. `1000/sqrt(3)` m
D. 1500 m

Answer

To solve the problem step by step, we will use the principles of trigonometry, particularly focusing on the angles of elevation and the properties of similar triangles.

### Step 1: Understand the Setup
We have two points A and B on a horizontal plane, which are 1000 meters apart. A balloon is above point A at one moment and above point B at another moment. The angles of elevation from B to the balloon above A is 60 degrees, and from A to the balloon above B is 30 degrees.

### Step 2: Draw the Diagram
Draw a horizontal line representing the ground with points A and B. Draw the balloon's positions above A and B. Label the height of the balloon above A as \( h_A \) and above B as \( h_B \).

### Step 3: Use Trigonometric Ratios
From point B, when the balloon is above A:
- The angle of elevation is 60 degrees.
- Using the tangent function:
\[ \tan(60^\circ) = \frac{h_A}{AB} \implies \sqrt{3} = \frac{h_A}{1000} \]
Therefore,
\[ h_A = 1000 \sqrt{3} \text{ meters} \]

From point A, when the balloon is above B:
- The angle of elevation is 30 degrees.
- Using the tangent function:
\[ \tan(30^\circ) = \frac{h_B}{AB} \implies \frac{1}{\sqrt{3}} = \frac{h_B}{1000} \]
Therefore,
\[ h_B = \frac{1000}{\sqrt{3}} \text{ meters} \]

### Step 4: Set Up Similar Triangles
The triangles formed by the points A, B, and the balloon positions are similar. The heights \( h_A \) and \( h_B \) correspond to the distances from A and B to the point where the balloon strikes the ground.

### Step 5: Use the Properties of Similar Triangles
From the similarity of triangles:
\[ \frac{h_A}{h_B} = \frac{AP}{BQ} \]
Where \( AP \) is the horizontal distance from A to the point where the balloon strikes the ground, and \( BQ \) is the horizontal distance from B to the point where the balloon strikes the ground.

### Step 6: Calculate the Distances
We have:
\[ h_A = 1000 \sqrt{3} \quad \text{and} \quad h_B = \frac{1000}{\sqrt{3}} \]
Let \( AP = x \) and \( BQ = 1000 - x \).

Using the ratio:
\[ \frac{1000 \sqrt{3}}{\frac{1000}{\sqrt{3}}} = \frac{x}{1000 - x} \]
Cross-multiplying gives:
\[ 1000 \sqrt{3} (1000 - x) = 1000 \cdot \sqrt{3} \cdot x \]
Simplifying:
\[ 1000 \sqrt{3} \cdot 1000 - 1000 \sqrt{3} \cdot x = 1000 \cdot \sqrt{3} \cdot x \]
\[ 1000 \sqrt{3} \cdot 1000 = 2000 \sqrt{3} \cdot x \]
\[ x = \frac{1000 \cdot 1000}{2000} = 500 \text{ meters} \]

### Final Step: Conclusion
The distance from point A to the point where the balloon strikes the plane is **500 meters**.