A flagstaff stands vertically on a pillar, the height of the flagstaff being double the height of the pillar. A man on the ground at a distance finds that both the pillar and the flagstaff subtend equal angles at his eyes. The ratio of the height of the pillar and the distance of the man from the pillar is

Options

A. `(sqrt3)/1`
B. `1/3`
C. `1/(sqrt3)`
D. `(sqrt3)/2`

Answer

To solve the problem, we need to find the ratio of the height of the pillar to the distance of the man from the pillar. Let's denote the height of the pillar as \( h \) and the distance of the man from the pillar as \( x \). The height of the flagstaff is given to be double the height of the pillar, so the height of the flagstaff is \( 2h \).

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- Let the height of the pillar be \( h \).
- Therefore, the height of the flagstaff is \( 2h \).
- The man is standing at a distance \( x \) from the base of the pillar.

2. **Setting Up the Angles**:
- The angle subtended by the pillar at the man's eye level is \( \theta \).
- The angle subtended by the flagstaff at the man's eye level is also \( \theta \) (as per the problem statement).

3. **Using Tangent Ratios**:
- For the pillar, we can write:
\[ \tan(\theta) = \frac{h}{x} \]
- For the flagstaff, we can write:
\[ \tan(2\theta) = \frac{2h}{x} \]

4. **Using the Double Angle Formula**:
- We know from trigonometry that:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
- Substituting \( \tan(\theta) = \frac{h}{x} \) into the double angle formula:
\[ \tan(2\theta) = \frac{2\left(\frac{h}{x}\right)}{1 - \left(\frac{h}{x}\right)^2} \]

5. **Equating the Two Expressions for \( \tan(2\theta) \)**:
- From step 3 and step 4, we have:
\[ \frac{2h}{x} = \frac{2\left(\frac{h}{x}\right)}{1 - \left(\frac{h}{x}\right)^2} \]

6. **Cross Multiplying**:
- Cross multiplying gives:
\[ 2h(1 - \left(\frac{h}{x}\right)^2) = 2h \]
- Simplifying this, we can cancel \( 2h \) (assuming \( h \neq 0 \)):
\[ 1 - \left(\frac{h}{x}\right)^2 = 1 \]
- This simplifies to:
\[ \left(\frac{h}{x}\right)^2 = \frac{1}{3} \]

7. **Finding the Ratio**:
- Taking the square root of both sides:
\[ \frac{h}{x} = \frac{1}{\sqrt{3}} \]
- Therefore, the ratio of the height of the pillar to the distance of the man from the pillar is:
\[ \frac{h}{x} = \frac{1}{\sqrt{3}} \]

### Final Answer:
The ratio of the height of the pillar to the distance of the man from the pillar is \( \frac{1}{\sqrt{3}} \).