Step-by-Step Solution
Key Concept: For a conic section, the slope of a tangent at a point can be expressed in terms of the coordinates and parameters of that point.
Given that the tangent with slope $m_3 = \frac{k}{2}$ passes through $(x_1, y_1)$, we have $m_3 = \frac{y_1}{x_1} \cdot 2$ from the condition. Therefore, $\frac{k}{2} = \frac{y_1}{x_1} \cdot 2$, which gives $m_3 = \frac{y_1}{2}$ as the slope of the tangent at the given point.
Correct Answer: 2