If the angles of elevation of the top of tower from three collinear points `A`, `B` and `C`, on a line leading to the foot of the tower, are `30^(@)`, `45^(@)` and `60^(@)` respectively, then the ratio , `AB : BC` is

Options

A. `sqrt(3) : 1`
B. `sqrt(3) : sqrt(2)`
C. `1 : sqrt(3)`
D. `2 : 3`

Answer

To solve the problem of finding the ratio \( AB : BC \) given the angles of elevation from points \( A \), \( B \), and \( C \) to the top of a tower, we can follow these steps:

### Step 1: Draw the Diagram
Draw a vertical tower \( OT \) where \( O \) is the foot of the tower and \( T \) is the top. Mark the points \( A \), \( B \), and \( C \) on the ground such that they are collinear and leading towards the tower. Label the angles of elevation from these points:
- Angle at \( C \) (60°)
- Angle at \( B \) (45°)
- Angle at \( A \) (30°)

### Step 2: Set Up the Relationships Using Trigonometry
Using the tangent function, we can express the height of the tower \( H \) in terms of the distances from the foot of the tower to each point:
1. From point \( C \):
\[ \tan(60^\circ) = \frac{H}{OC} \implies H = OC \cdot \sqrt{3} \]
2. From point \( B \):
\[ \tan(45^\circ) = \frac{H}{OB} \implies H = OB \implies H = OC + BC \]
3. From point \( A \):
\[ \tan(30^\circ) = \frac{H}{OA} \implies H = OA \cdot \frac{1}{\sqrt{3}} \implies H = OC + BC + AB \cdot \frac{1}{\sqrt{3}} \]

### Step 3: Substitute and Rearrange
Now we have three equations for \( H \):
1. \( H = OC \cdot \sqrt{3} \)
2. \( H = OC + BC \)
3. \( H = OC + BC + AB \cdot \frac{1}{\sqrt{3}} \)

### Step 4: Equate the Expressions for \( H \)
From the first two equations:
\[ OC \cdot \sqrt{3} = OC + BC \]
Rearranging gives:
\[ OC(\sqrt{3} - 1) = BC \quad \text{(1)} \]

From the second and third equations:
\[ OC + BC = OC + BC + AB \cdot \frac{1}{\sqrt{3}} \]
This simplifies to:
\[ 0 = AB \cdot \frac{1}{\sqrt{3}} \quad \text{(2)} \]

### Step 5: Solve for \( AB \) and \( BC \)
From equation (1):
\[ BC = OC(\sqrt{3} - 1) \]

From equation (2), we can express \( AB \) in terms of \( BC \):
\[ AB = BC \cdot \left(\sqrt{3} - 1\right) \]

### Step 6: Find the Ratio \( AB : BC \)
Now we can find the ratio:
\[ \frac{AB}{BC} = \frac{BC \cdot \left(\sqrt{3} - 1\right)}{BC} = \sqrt{3} - 1 \]
Thus, the ratio \( AB : BC \) is:
\[ AB : BC = \sqrt{3} : 1 \]

### Final Answer
The required ratio \( AB : BC \) is \( \sqrt{3} : 1 \).

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