The auxiliary circle of a family of ellipses passes through the origin and makes intercepts of 8 units and 6 units on the x and y-axis, respectively. If the eccentricity of all such ellipses is `1/2,` then find the locus of the focus.

Explanation

To find the locus of the focus of the family of ellipses, we can follow these steps:

### Step 1: Understand the Problem
The auxiliary circle of the family of ellipses passes through the origin and makes intercepts of 8 units on the x-axis and 6 units on the y-axis. The eccentricity of the ellipses is given as \( \frac{1}{2} \).

### Step 2: Determine the Center of the Auxiliary Circle
The intercepts on the axes give us the points:
- On the x-axis: \( (8, 0) \)
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