If two tangents are drawn from a point on `x^(2)+y^(2)=16` to the circle `x^(2)+y^(2)=8` then the angle between the tangents is

Options

A. `(pi)/2`
B. `(pi)/4`
C. `(2pi)/3`
D. `(pi)`

Answer

To solve the problem of finding the angle between the tangents drawn from a point on the circle \( x^2 + y^2 = 16 \) to the circle \( x^2 + y^2 = 8 \), we can follow these steps:

### Step 1: Identify the circles and their properties
The first circle \( C_1 \) has the equation:
\[ x^2 + y^2 = 8 \]
This can be rewritten in the standard form as:
\[ x^2 + y^2 = r_1^2 \]
where \( r_1 = \sqrt{8} = 2\sqrt{2} \).

The second circle \( C_2 \) has the equation:
\[ x^2 + y^2 = 16 \]
This can be rewritten in the standard form as:
\[ x^2 + y^2 = r_2^2 \]
where \( r_2 = \sqrt{16} = 4 \).

### Step 2: Check if \( C_2 \) is the director circle of \( C_1 \)
For \( C_2 \) to be the director circle of \( C_1 \), the following condition must hold:
\[ r_2 = \sqrt{2} \cdot r_1 \]
Substituting the values of \( r_1 \) and \( r_2 \):
\[ 4 = \sqrt{2} \cdot (2\sqrt{2}) \]
Calculating the right-hand side:
\[ \sqrt{2} \cdot (2\sqrt{2}) = 2 \cdot 2 = 4 \]
Since both sides are equal, \( C_2 \) is indeed the director circle of \( C_1 \).

### Step 3: Determine the angle between the tangents
When a circle is a director circle of another circle, the angle between the tangents drawn from a point on the outer circle (in this case, \( C_2 \)) to the inner circle (in this case, \( C_1 \)) is \( \frac{\pi}{2} \) radians (or 90 degrees).

### Conclusion
Thus, the angle between the tangents drawn from a point on the circle \( x^2 + y^2 = 16 \) to the circle \( x^2 + y^2 = 8 \) is:
\[ \frac{\pi}{2} \]