From any point on the circle `x^(2)+y^(2)=a^(2)` tangents are drawn to the circle `x^(2)+y^(2)=a^(2) sin^(2) theta`. The angle between them is

Options

A. `alpha//2`
B. `alpha`
C. `2alpha`
D. `4alpha`

Answer

To solve the problem, we need to find the angle between the two tangents drawn from any point on the circle \(x^2 + y^2 = a^2\) to the circle \(x^2 + y^2 = a^2 \sin^2 \theta\).

### Step-by-Step Solution:

1. **Identify the Circles**:
- The first circle is given by the equation \(x^2 + y^2 = a^2\).
- Center: \(O(0, 0)\)
- Radius: \(R_1 = a\)
- The second circle is given by the equation \(x^2 + y^2 = a^2 \sin^2 \theta\).
- Center: \(O(0, 0)\)
- Radius: \(R_2 = a \sin \theta\)

2. **Understanding the Tangents**:
- From any point \(P\) on the first circle, two tangents can be drawn to the second circle. The angle between these tangents is what we need to find.

3. **Using Geometry**:
- The angle between the two tangents drawn from a point \(P\) to a circle can be found using the relationship between the angles and the radii of the circles.
- Let the angle between the two tangents be \(2\alpha\).

4. **Forming a Right Triangle**:
- Consider the right triangle formed by the radius of the second circle \(R_2\) (which is \(a \sin \theta\)), the radius of the first circle \(R_1\) (which is \(a\)), and the line segment connecting the center of the circles to the point \(P\).
- The angle \( \alpha \) is formed at point \(P\) between the two tangents.

5. **Applying the Sine Rule**:
- In triangle \(OPQ\) (where \(O\) is the center, \(P\) is the point on the first circle, and \(Q\) is the point where the tangent touches the second circle):
\[ \sin \alpha = \frac{R_2}{R_1} = \frac{a \sin \theta}{a} = \sin \theta \]
- Thus, we have:
\[ \alpha = \theta \]

6. **Finding the Angle Between the Tangents**:
- Since the angle between the two tangents is \(2\alpha\), we can substitute the value of \(\alpha\):
\[ 2\alpha = 2\theta \]

### Final Answer:
The angle between the two tangents is \(2\theta\).

---