This problem combines the concept of tangents to a hyperbola, specifically using the slope form, with careful analysis of x-intercepts and coordinate geometry. We will break down the solution into clear, manageable steps.

Key Concept: Tangent Equation of a Hyperbola in Slope Form For a hyperbola of the form $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ (transverse axis along the y-axis), the equation of a tangent with slope $m$ is given by: $y = mx \pm \sqrt{a^2m^2 - b^2}$ This formula is critical for solving the problem. Note that the sign before the square root depends on the specific tangent and