The angle between the pair of tangents drawn from the point (2,4) to the circle `x^(2)+y^(2)=4` is

Options

A. `tan^(-1)(3//8)`
B. `tan^(-1)(4//3)`
C. `90^(@)`
D. none

Answer

To find the angle between the pair of tangents drawn from the point (2, 4) to the circle defined by the equation \( x^2 + y^2 = 4 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Circle's Center and Radius**:
The given circle is \( x^2 + y^2 = 4 \).
- Center: \( (0, 0) \)
- Radius: \( r = \sqrt{4} = 2 \)

2. **Identify the Point from which Tangents are Drawn**:
The point from which tangents are drawn is \( P(2, 4) \).

3. **Calculate the Distance from the Point to the Center of the Circle**:
The distance \( d \) from point \( P(2, 4) \) to the center \( O(0, 0) \) is calculated using the distance formula:
\[ d = \sqrt{(2 - 0)^2 + (4 - 0)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \]

4. **Use the Formula for the Angle Between the Tangents**:
The angle \( \theta \) between the two tangents from point \( P \) to the circle can be calculated using the formula:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{r}{d} \]
Substituting the values:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{2}{2\sqrt{5}} = \frac{1}{\sqrt{5}} \]

5. **Calculate \( \theta \)**:
To find \( \theta \), we first find \( \frac{\theta}{2} \):
\[ \frac{\theta}{2} = \tan^{-1}\left(\frac{1}{\sqrt{5}}\right) \]
Therefore,
\[ \theta = 2 \tan^{-1}\left(\frac{1}{\sqrt{5}}\right) \]

6. **Final Result**:
The angle between the pair of tangents drawn from the point \( (2, 4) \) to the circle \( x^2 + y^2 = 4 \) is:
\[ \theta = 2 \tan^{-1}\left(\frac{1}{\sqrt{5}}\right) \]