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)alpha`. The angle between them is

Options

A. `alpha//2`
B. `alpha`
C. `2alpha`
D. none of these

Answer

To solve the problem of finding the angle between the tangents drawn from any point on the circle \(x^2 + y^2 = a^2\) to the circle \(x^2 + y^2 = a^2 \sin^2 \alpha\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Circles**:
- The first circle is given by the equation \(x^2 + y^2 = a^2\). This circle has a center at the origin \((0, 0)\) and a radius \(R_1 = a\).
- The second circle is given by the equation \(x^2 + y^2 = a^2 \sin^2 \alpha\). This circle also has a center at the origin and a radius \(R_2 = a \sin \alpha\).

2. **Draw Tangents**:
- From any point \(P\) on the first circle, tangents are drawn to the second circle. Let the points where the tangents touch the second circle be \(T_1\) and \(T_2\).

3. **Use the Angle Between Tangents Formula**:
- The angle \(\theta\) between the two tangents drawn from a point \(P\) to a circle can be found using the formula:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{R_2}{d} \]
where \(R_2\) is the radius of the second circle, and \(d\) is the distance from the point \(P\) to the center of the second circle.

4. **Calculate the Distance \(d\)**:
- Since point \(P\) lies on the first circle, the distance \(d\) from the origin (the center of both circles) to point \(P\) is equal to the radius of the first circle, which is \(a\).

5. **Substitute Values**:
- Now, substituting the values into the tangent formula:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{a \sin \alpha}{a} = \sin \alpha \]

6. **Find \(\theta\)**:
- To find \(\theta\), we can use the double angle identity for tangent:
\[ \tan(\theta) = \frac{2 \tan\left(\frac{\theta}{2}\right)}{1 - \tan^2\left(\frac{\theta}{2}\right)} \]
- Since \(\tan\left(\frac{\theta}{2}\right) = \sin \alpha\), we can substitute:
\[ \tan(\theta) = \frac{2 \sin \alpha}{1 - \sin^2 \alpha} = \frac{2 \sin \alpha}{\cos^2 \alpha} \]
- Thus, \(\theta = 2\alpha\).

7. **Conclusion**:
- Therefore, the angle between the tangents drawn from any point on the first circle to the second circle is \(2\alpha\).

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