From two points on the ground and lying on a straight line through the foot of a pillar, the two angles of elevation of the top of the pillar are complementary to each other. If the distances of the two points from the foot of the pillar are 12 metres and 27 metres and the two points lie on the same side of the pillar, then the height (in metres) of the pillar is:

Options

A. 16
B. 12
C. 15
D. 18

Answer

To solve the problem step by step, we will denote the height of the pillar as \( h \).

### Step 1: Understand the problem
We have two points on the ground at distances of 12 meters and 27 meters from the foot of the pillar. The angles of elevation from these points to the top of the pillar are complementary, meaning they add up to 90 degrees.

### Step 2: Set up the triangles
Let:
- Point A be 12 meters from the foot of the pillar.
- Point B be 27 meters from the foot of the pillar.
- The height of the pillar be \( h \).

From point A, the angle of elevation to the top of the pillar is \( \theta \).
From point B, the angle of elevation to the top of the pillar is \( 90^\circ - \theta \).

### Step 3: Use the tangent function
Using the definition of tangent in right triangles:
1. From point A:
\[ \tan(\theta) = \frac{h}{12} \]
Therefore, we can express \( h \) as:
\[ h = 12 \tan(\theta) \quad \text{(1)} \]

2. From point B:
\[ \tan(90^\circ - \theta) = \cot(\theta) = \frac{h}{27} \]
Therefore, we can express \( h \) as:
\[ h = 27 \cot(\theta) \quad \text{(2)} \]

### Step 4: Relate the two expressions for \( h \)
From equations (1) and (2), we can set them equal to each other:
\[ 12 \tan(\theta) = 27 \cot(\theta) \]

### Step 5: Use the identity for cotangent
Recall that \( \cot(\theta) = \frac{1}{\tan(\theta)} \). Substituting this into the equation gives:
\[ 12 \tan(\theta) = 27 \cdot \frac{1}{\tan(\theta)} \]

### Step 6: Multiply both sides by \( \tan(\theta) \)
\[ 12 \tan^2(\theta) = 27 \]

### Step 7: Solve for \( \tan^2(\theta) \)
\[ \tan^2(\theta) = \frac{27}{12} = \frac{9}{4} \]

### Step 8: Take the square root
\[ \tan(\theta) = \frac{3}{2} \]

### Step 9: Substitute back to find \( h \)
Using equation (1):
\[ h = 12 \tan(\theta) = 12 \cdot \frac{3}{2} = 18 \]

### Conclusion
The height of the pillar is \( \boxed{18} \) meters.