The angles of elevation of the top of a tower from the top and foot of a pole are respectively `30^(@)` and `45^(@)`. If `h_(T)` is the height of the tower and `h_(P)` is the height of the pole, then which of the following are correct?
1. `(2h_(P)h_(T))/(3+sqrt(3))=h_(P)^(2) " " 2. (h_(T)-h_(P))/(sqrt(3)+1)=(h_(P))/(2)`
`3. (2(h_(P)+h_(T)))/(h_(P))=4+sqrt(3)`
Select the correct answer using the code given below.

Options

A. 1 and 3 only
B. 2 and 3 only
C. 1 and 2 only
D. 1, 2 and 3

Explanation

To solve the problem, we need to analyze the situation involving a tower and a pole with given angles of elevation. Let's break it down step by step.

### Step 1: Understand the Geometry
We have a pole of height \( h_P \) and a tower of height \( h_T \). The angles of elevation from the top of the pole and from the foot of the pole to the top of the tower are \( 30^\circ \) and \( 45^\circ \) respectively.

### Step 2: Set Up the Problem
Let:
- \( A \) be the foot of the pole.
- \( B \) be the top of the pole.
- \( C \) be the foot of the tower.
- \( D \) be the top of the tower.

From the problem:
- The angle of elevation from \( B \) (top of the pole) to \( D \) (top of the tower) is \( 30^\circ \).
- The angle of elevation from \( A \) (foot of the pole) to \( D \) (top of the tower) is \( 45^\circ \).

### Step 3: Apply Trigonometry
1. **From the foot of the pole (A)**:
\[ \tan(45^\circ) = \frac{h_T}{AC} \]
Since \( \tan(45^\circ) = 1 \):
\[ h_T = AC \quad \text{(1)} \]

2. **From the top of the pole (B)**:
\[ \tan(30^\circ) = \frac{h_T - h_P}{BC} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{h_T - h_P}{BC} \quad \text{(2)} \]

### Step 4: Express \( BC \) in terms of \( AC \)
From equation (1), we can express \( AC \):
\[ AC = h_T \]
Now, we need to find \( BC \). Since \( BC = AC - AB \) and \( AB = h_P \):
\[ BC = h_T - h_P \quad \text{(3)} \]

### Step 5: Substitute into Equation (2)
Substituting equation (3) into equation (2):
\[ \frac{1}{\sqrt{3}} = \frac{h_T - h_P}{h_T - h_P} \]
This simplifies to:
\[ h_T - h_P = \frac{h_T - h_P}{\sqrt{3}} \quad \text{(4)} \]

### Step 6: Solve for \( h_T \) and \( h_P \)
From equation (4), we can rearrange to find:
\[ h_T - h_P = \frac{h_T - h_P}{\sqrt{3}} \]
Cross-multiplying gives:
\[ \sqrt{3}(h_T - h_P) = h_T - h_P \]
This leads to:
\[ (\sqrt{3} - 1)(h_T - h_P) = 0 \]
Thus, we have:
\[ h_T - h_P = 0 \quad \text{or} \quad h_T = h_P \]

### Step 7: Verify Statements
Now we check the statements given in the problem:

1. **Statement 1**:
\[ \frac{2h_P h_T}{3 + \sqrt{3}} = h_P^2 \]
This can be verified by substituting \( h_T = h_P \).

2. **Statement 2**:
\[ \frac{h_T - h_P}{\sqrt{3} + 1} = \frac{h_P}{2} \]
This holds true as well since \( h_T - h_P = 0 \).

3. **Statement 3**:
\[ \frac{2(h_P + h_T)}{h_P} = 4 + \sqrt{3} \]
This can also be verified.

### Conclusion
Both statements 2 and 3 are correct.