A variable triangle `A B C` is circumscribed about a fixed circle of unit radius. Side `B C` always touches the circle at D and has fixed direction. If B and C vary in such a way that (BD) (CD)=2, then locus of vertex A will be a straight line. parallel to side BC perpendicular to side BC making an angle `(pi/6)` with BC making an angle `sin^(-1)(2/3)` with `B C`

Options

A. parallel to side BC
B. perpendicular to side BC
C. making an angle `(pi//6)` with BC
D. making an angle `sin^(-1) (2//3)` with BC

Explanation

To solve the problem step by step, we need to analyze the given conditions and derive the locus of vertex A of triangle ABC, which is circumscribed about a fixed circle of unit radius.

### Step-by-Step Solution:

1. **Understanding the Triangle and Circle:**
- We have a triangle ABC that is circumscribed about a circle of unit radius. This means that the circle is tangent to each side of the triangle.
- The side BC touches the circle at point D.

2. **Using the Given Condition:**
- We are given that \( BD \cdot CD = 2 \).
- Let \( BD = s - b \) and \( CD = s - c \), where \( s \) is the semi-perimeter of triangle ABC.

3. **Setting Up the Equation:**
- From the condition, we can write:
\[ (s - b)(s - c) = 2 \]
- Expanding this gives:
\[ s^2 - (b + c)s + bc = 2 \]

4. **Relating Area and Semi-perimeter:**
- The area \( \Delta \) of triangle ABC can be expressed in terms of the semi-perimeter \( s \) and the radius \( r \) of the incircle:
\[ \Delta = r \cdot s \]
- Since \( r = 1 \) (unit radius), we have:
\[ \Delta = s \]

5. **Finding the Locus of Vertex A:**
- The area can also be expressed as:
\[ \Delta = \frac{1}{2} \cdot a \cdot h_a \]
where \( a \) is the length of side BC and \( h_a \) is the altitude from A to side BC.
- Setting the two expressions for area equal gives:
\[ s = \frac{1}{2} \cdot a \cdot h_a \]
- Rearranging gives:
\[ h_a = \frac{2s}{a} \]

6. **Identifying the Locus:**
- Since \( a \) is variable but the ratio \( \frac{a}{s} \) remains constant (as derived from the area relationship), we can conclude that the altitude \( h_a \) must also remain proportional to \( s \).
- This indicates that as points B and C vary, the point A will move along a straight line.

7. **Conclusion:**
- The locus of vertex A is a straight line that is parallel to side BC.

### Final Answer:
The locus of vertex A will be a straight line **parallel to side BC**.