If two tangents are drawn from a point to the circle `x^(2) + y^(2) =32` to the circle `x^(2) + y^(2) = 16`, then the angle between the tangents is

Options

A. `(-pi)/(4)`
B. `(pi)/(3)`
C. `(pi)/(2)`
D. `(pi)/(6)`

Answer

To find the angle between the tangents drawn from a point to the two circles given by the equations \( x^2 + y^2 = 32 \) and \( x^2 + y^2 = 16 \), we can follow these steps:

### Step 1: Identify the radii of the circles
The equations of the circles can be rewritten in the standard form \( x^2 + y^2 = r^2 \).

- For the first circle \( x^2 + y^2 = 32 \):
\[ r_1 = \sqrt{32} = 4\sqrt{2} \]

- For the second circle \( x^2 + y^2 = 16 \):
\[ r_2 = \sqrt{16} = 4 \]

### Step 2: Establish the relationship between the radii
We can observe that:
\[ r_1 = 4\sqrt{2} \quad \text{and} \quad r_2 = 4 \]
We can express \( r_1 \) in terms of \( r_2 \):
\[ r_1 = \sqrt{2} \cdot r_2 \]

### Step 3: Use the property of tangents
There is a known property regarding the angle \( \theta \) between the tangents drawn from a point to two circles with radii \( r_1 \) and \( r_2 \):
\[ \tan\left(\frac{\theta}{2}\right) = \frac{r_1 - r_2}{r_1 + r_2} \]

### Step 4: Calculate the angle between the tangents
Substituting the values of \( r_1 \) and \( r_2 \):
\[ \tan\left(\frac{\theta}{2}\right) = \frac{4\sqrt{2} - 4}{4\sqrt{2} + 4} \]
Simplifying this expression:
\[ = \frac{4(\sqrt{2} - 1)}{4(\sqrt{2} + 1)} = \frac{\sqrt{2} - 1}{\sqrt{2} + 1} \]

### Step 5: Determine the angle \( \theta \)
Since \( r_1 = \sqrt{2} \cdot r_2 \), we can use the property that if \( r_1 = k \cdot r_2 \) where \( k = \sqrt{2} \), then the angle \( \theta \) between the tangents is:
\[ \theta = \frac{\pi}{2} \]

### Final Answer
Thus, the angle between the tangents drawn from a point to the two circles is:
\[ \theta = \frac{\pi}{2} \text{ radians} \]

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