This problem requires us to find the locus of the midpoints of line segments formed by the intersection of a tangent to a parabola and a hyperbola. The locus is stated to be a parabola, and we need to determine its properties. The core strategy involves expressing the same line (the tangent, which acts as a chord) in two different forms and then equating the coefficients to find the relationship between the coordinates of the midpoint.

1. Key Concepts and Formulas

We will utilize two fundamental formulas from coordinate geometry:

• Equation of a Tangent to a Parabola in Slope Form: For a parabola of the form $y^2 = 4ax$, the equation of a tangent with slope $m$ is given by $$y = mx + \frac{a}{