1. Understanding the Problem and Key Concepts

The problem asks us to find the locus of the midpoint of the line segment AB, where A and B are the points where a tangent to the given hyperbola intersects the x and y axes, respectively. This is a classic locus problem involving tangents to conic sections.

To solve this, we will use the following key concepts and formulas:

• Standard Equation of a Hyperbola: A hyperbola centered at the origin can be written in the form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ or $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$.
• **Equation of