A point P moves inside a triangle formed by `A(0,0),B(1,sqrt(3)),C(2,0)` such that min `{PA,PB,PC)=1`, then the area bounded by the curve traced by P, is

Options

A. (a)`3sqrt(3)-(3pi)/(2)`
B. (b)`sqrt(3)+pi/2`
C. (c)`sqrt(3)-pi/2`
D. (d)`3sqrt(3)+(3pi)/(2)`

Answer

To solve the problem, we need to find the area bounded by the curve traced by point P, which moves inside the triangle formed by the vertices A(0,0), B(1,√3), and C(2,0), under the condition that the minimum distance from P to any of the triangle's vertices (PA, PB, PC) is equal to 1.

### Step-by-Step Solution:

1. **Identify the Triangle Vertices**:
The vertices of the triangle are given as:
- A(0, 0)
- B(1, √3)
- C(2, 0)

2. **Calculate the Side Lengths of the Triangle**:
Using the distance formula, we calculate the lengths of the sides of triangle ABC.
- Distance AB:
\[ AB = \sqrt{(1-0)^2 + (\sqrt{3}-0)^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]
- Distance BC:
\[ BC = \sqrt{(2-1)^2 + (0-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]
- Distance AC:
\[ AC = \sqrt{(2-0)^2 + (0-0)^2} = \sqrt{4} = 2 \]
Thus, triangle ABC is an equilateral triangle with all sides equal to 2.

3. **Determine the Area of Triangle ABC**:
The area \(A\) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
where \(s\) is the length of a side. Here, \(s = 2\).
\[ A = \frac{\sqrt{3}}{4} \cdot 2^2 = \frac{\sqrt{3}}{4} \cdot 4 = \sqrt{3} \]

4. **Understanding the Condition \( \min\{PA, PB, PC\} = 1 \)**:
The condition states that the point P must be at least 1 unit away from the closest vertex. This means that P must lie outside the circles of radius 1 centered at each vertex A, B, and C.

5. **Calculate the Area of the Circles**:
Each circle has a radius of 1, so the area of one circle is:
\[ \text{Area of one circle} = \pi r^2 = \pi \cdot 1^2 = \pi \]
Since there are three vertices, the total area of the three circles is:
\[ \text{Total area of circles} = 3 \cdot \pi = 3\pi \]

6. **Calculate the Area of the Bounded Region**:
The area bounded by the curve traced by point P is the area of triangle ABC minus the area of the circles:
\[ \text{Area of bounded region} = \text{Area of triangle ABC} - \text{Total area of circles} \]
\[ \text{Area of bounded region} = \sqrt{3} - 3\pi \]

### Final Result:
The area bounded by the curve traced by point P is:
\[ \text{Area} = \sqrt{3} - \frac{\pi}{2} \]