The angle of elevation of the top of a tower from the point P and Q at distance of a and b respectively from the base of the tower and in the same straight line with it are complementary .The height of the tower is

Options

A. `sqrt(ab)`
B. `(a)/(b)`
C. ab
D. `a^(2)b^(2)`

Answer

To solve the problem of finding the height of the tower given the complementary angles of elevation from points P and Q, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Let the height of the tower be \( h \).
- The distance from point P to the base of the tower is \( a \).
- The distance from point Q to the base of the tower is \( b \).
- The angles of elevation from points P and Q to the top of the tower are complementary. This means if the angle from P is \( \theta \), then the angle from Q is \( 90^\circ - \theta \).

2. **Setting Up the Triangles**:
- From point P, we can use the tangent function:
\[ \tan(\theta) = \frac{h}{a} \]
- From point Q, using the complementary angle:
\[ \tan(90^\circ - \theta) = \cot(\theta) = \frac{h}{b} \]

3. **Using the Cotangent Identity**:
- We know that \( \cot(\theta) = \frac{1}{\tan(\theta)} \). Therefore, we can express the second equation as:
\[ \cot(\theta) = \frac{b}{h} \]
- This implies:
\[ \tan(\theta) = \frac{h}{a} \quad \text{and} \quad \cot(\theta) = \frac{b}{h} \]

4. **Relating the Two Equations**:
- From the first equation, we can express \( h \):
\[ h = a \tan(\theta) \]
- From the second equation, we can express \( h \) in terms of \( b \):
\[ h = \frac{b}{\cot(\theta)} = b \tan(\theta) \]

5. **Equating the Two Expressions for Height**:
- Since both expressions equal \( h \):
\[ a \tan(\theta) = b \tan(\theta) \]
- Dividing both sides by \( \tan(\theta) \) (assuming \( \tan(\theta) \neq 0 \)):
\[ a = b \]

6. **Finding the Height**:
- We can now express \( h \) in terms of \( a \) and \( b \):
\[ h = \sqrt{ab} \]

### Final Answer:
The height of the tower is given by:
\[ h = \sqrt{ab} \]