Two poles of height a and b stand at the centers of two circular plots which touch each other externally at a point and the two poles subtend angles of 30° and 60° respectively at this point, then distance between the centers of these plots is

Options

A. `a+b`
B. `((3a+b))/(sqrt3)`
C. `((a+3b))/(sqrt3)`
D. `asqrt3+b`

Answer

To solve the problem, we need to find the distance between the centers of two circular plots that touch each other externally, given the heights of the poles and the angles they subtend at the point of contact.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- Let the height of the first pole be \( a \) and the height of the second pole be \( b \).
- The first pole subtends an angle of \( 30^\circ \) and the second pole subtends an angle of \( 60^\circ \) at the point where the two circular plots touch.

2. **Setting Up the Triangles**:
- From the point of contact, draw perpendiculars to the heights of the poles. This creates two right triangles:
- Triangle formed by the first pole (height \( a \)) and the radius of the first circular plot \( r_1 \).
- Triangle formed by the second pole (height \( b \)) and the radius of the second circular plot \( r_2 \).

3. **Using Trigonometric Ratios**:
- For the first triangle (angle \( 30^\circ \)):
\[ \tan(30^\circ) = \frac{a}{r_1} \]
Knowing that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), we have:
\[ \frac{1}{\sqrt{3}} = \frac{a}{r_1} \implies r_1 = \sqrt{3} a \]

- For the second triangle (angle \( 60^\circ \)):
\[ \tan(60^\circ) = \frac{b}{r_2} \]
Knowing that \( \tan(60^\circ) = \sqrt{3} \), we have:
\[ \sqrt{3} = \frac{b}{r_2} \implies r_2 = \frac{b}{\sqrt{3}} \]

4. **Finding the Distance Between the Centers**:
- The distance between the centers of the two circular plots is the sum of the two radii:
\[ d = r_1 + r_2 \]
- Substituting the values of \( r_1 \) and \( r_2 \):
\[ d = \sqrt{3} a + \frac{b}{\sqrt{3}} \]

5. **Combining the Terms**:
- To combine the terms, we can express them with a common denominator:
\[ d = \frac{3a + b}{\sqrt{3}} \]

### Final Answer:
Thus, the distance between the centers of the two circular plots is:
\[ \frac{3a + b}{\sqrt{3}} \]