Step 1: Understanding the Question:
The tangents are drawn from an external point to a circle. The locus will be a concentric circle. We need to find its radius.
Step 2: Key Formula or Approach:
Let the external point be \( P(h, k) \). If \( \theta \) is the angle between tangents, then \( \sin(\theta/2) = \frac{r}{\text{OP}} \), where \( r \) is the radius of the circle and \( \text{OP} \) is the distance from center to \( P \).
Step 3: Detailed Explanation:
Given circle: \( x^2 + y^2 = 4^2 \Rightarrow r = 4 \). Center \( O = (0, 0) \).
Angle \( \theta = 60^\circ \Rightarrow \theta/2 = 30^\circ \).
\[ \sin 30^\circ = \frac{4}{\sqrt{h^2 + k^2}} \]
\[ \frac{1}{2} = \frac{4}{\sqrt{h^2 + k^2}} \]
\[ \sqrt{h^2 + k^2} = 8 \Rightarrow h^2 + k^2 = 64 \]
Replacing \( (h, k) \) with \( (x, y) \), the locus is \( x^2 + y^2 = 64 \).
Step 4: Final Answer:
The locus is \( x^2 + y^2 = 64 \).