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 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 circles and their properties
The first circle is given by the equation:
\[ x^2 + y^2 = 32 \]
This circle has a center at \((0, 0)\) and a radius of \(\sqrt{32} = 4\sqrt{2}\).

The second circle is given by the equation:
\[ x^2 + y^2 = 16 \]
This circle also has a center at \((0, 0)\) and a radius of \(\sqrt{16} = 4\).

### Step 2: Determine the director circle
The director circle of a given circle is defined as the locus of points from which tangents drawn to the circle subtend a right angle (90 degrees). The radius of the director circle is given by:
\[ R_d = \sqrt{2} \times R \]
where \(R\) is the radius of the original circle.

For the circle \(x^2 + y^2 = 16\) (radius \(R = 4\)):
\[ R_d = \sqrt{2} \times 4 = 4\sqrt{2} \]

### Step 3: Compare the two circles
We have:
- The radius of the first circle \(R_1 = 4\sqrt{2}\) (for \(x^2 + y^2 = 32\))
- The radius of the second circle \(R_2 = 4\) (for \(x^2 + y^2 = 16\))

Since the radius of the first circle (the larger circle) is equal to the radius of the director circle of the second circle, we can conclude that the first circle is indeed the director circle of the second circle.

### Step 4: Conclusion about the angle between tangents
Since the first circle is the director circle of the second circle, the angle between the tangents drawn from any external point to the second circle will be \(90^\circ\) or \(\frac{\pi}{2}\) radians.

Thus, the angle between the tangents is:
\[ \boxed{\frac{\pi}{2}} \]

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