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:sqrt3`
C. `2:sqrt3`
D. `1:sqrt2`

Answer

To solve the problem step by step, we will denote the height of the pillar as \( h \) and the height of the flagstaff as \( 2h \) (since it is double the height of the pillar). Let the distance of the man from the base of the pillar be \( x \).

### Step 1: Set up the triangles
The man observes both the pillar and the flagstaff, and they subtend equal angles at his eyes. This means that the angle subtended by the height of the pillar \( h \) at distance \( x \) is equal to the angle subtended by the total height of the flagstaff and pillar \( (h + 2h = 3h) \) at the same distance \( x \).

### Step 2: Write the tangent ratios
Using the tangent function, we can express the angles:
- For the pillar:
\[ \tan(\theta) = \frac{h}{x} \]
- For the flagstaff:
\[ \tan(2\theta) = \frac{3h}{x} \]

### Step 3: Use the double angle formula for tangent
Using the double angle formula for tangent, we have:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
Substituting \( \tan(\theta) = \frac{h}{x} \) into the equation gives:
\[ \tan(2\theta) = \frac{2\left(\frac{h}{x}\right)}{1 - \left(\frac{h}{x}\right)^2} \]

### Step 4: Set the two expressions for \(\tan(2\theta)\) equal
Now we can set the two expressions for \(\tan(2\theta)\) equal to each other:
\[ \frac{3h}{x} = \frac{2\left(\frac{h}{x}\right)}{1 - \left(\frac{h}{x}\right)^2} \]

### Step 5: Cross-multiply and simplify
Cross-multiplying gives:
\[ 3h(1 - \left(\frac{h}{x}\right)^2) = 2\frac{h}{x} \cdot x \]
This simplifies to:
\[ 3h - 3\frac{h^3}{x^2} = 2h \]
Rearranging gives:
\[ 3h - 2h = 3\frac{h^3}{x^2} \]
\[ h = 3\frac{h^3}{x^2} \]

### Step 6: Solve for the ratio \( \frac{h}{x} \)
Dividing both sides by \( h \) (assuming \( h \neq 0 \)):
\[ 1 = 3\frac{h^2}{x^2} \]
This implies:
\[ \frac{h^2}{x^2} = \frac{1}{3} \]
Taking the square root of both sides gives:
\[ \frac{h}{x} = \frac{1}{\sqrt{3}} \]

### Final Answer
Thus, the ratio of the height of the pillar to the distance of the man from the pillar is:
\[ \frac{h}{x} = \frac{1}{\sqrt{3}} \]