Key Concepts and Formulas

To solve this problem effectively, we will utilize several fundamental concepts from coordinate geometry:

1. Intersection of Curves: To find the points where two curves intersect, we must find the $(x, y)$ coordinates that satisfy both of their equations simultaneously. This typically involves substitution or elimination methods.
2. Standard Parabola Equation: The given parabola $y^2 = 4x$ is in the standard form $y^2 = 4ax$. By comparing these, we can identify the parameter $a$. In this case, $4a = 4$, so $a = 1$. This parameter $a$ is crucial for writing the tangent equation.
3. Equation of Tangent to a Parabola: For