From two points on the ground lying on a straight line through the foot of a pillaer , the two angles of elevation of the top of the pillar are complementary to each other . If the distance of the two points from the foot of the pillar are 9 metres and 16 metres and the two points lie on the same side of the pillar , then the height of the pillar is

Options

A. 5 m
B. 10 m
C. 7 m
D. 12 m

Answer

To find the height of the pillar given the distances from two points and the fact that the angles of elevation are complementary, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a pillar with height \( h \). There are two points, \( P \) and \( Q \), on the ground from which the angles of elevation to the top of the pillar are complementary. The distances from the foot of the pillar to points \( P \) and \( Q \) are 9 meters and 16 meters, respectively.

2. **Define Angles**:
Let the angle of elevation from point \( P \) be \( \theta \). Therefore, the angle of elevation from point \( Q \) will be \( 90^\circ - \theta \) since they are complementary.

3. **Set Up the Tangent Ratios**:
From point \( P \):
\[ \tan(\theta) = \frac{h}{9} \quad \text{(1)} \]
From point \( Q \):
\[ \tan(90^\circ - \theta) = \cot(\theta) = \frac{h}{16} \quad \text{(2)} \]

4. **Relate Tangent and Cotangent**:
Recall that \( \cot(\theta) = \frac{1}{\tan(\theta)} \). Therefore, from equation (1):
\[ \cot(\theta) = \frac{9}{h} \]
Substituting this into equation (2):
\[ \frac{h}{16} = \frac{9}{h} \]

5. **Cross-Multiply**:
Cross-multiplying gives:
\[ h^2 = 9 \times 16 \]

6. **Calculate \( h^2 \)**:
\[ h^2 = 144 \]

7. **Find \( h \)**:
Taking the square root of both sides:
\[ h = \sqrt{144} = 12 \text{ meters} \]

### Conclusion:
The height of the pillar is \( 12 \) meters.