In this article, we present a new topological index based on a geometric approach. A new insight on the mathematical background of the hyperbolic directrix of classical conic geometry is used in development of the new topological index. When we consider an active edge as a coordinate point \( P(d(u), d(v)) \) with \( d(u) \le d(v) \), the usual Sombor index is developed by considering Euclidean distance to the origin, \( \sqrt{d(u)^2 + d(v)^2} \). Meanwhile the smaller degree \( d(u) \) measures the perpendicular distance from \( P \) to the vertical axis which acts as a structural directrix line. This defines the local edge contribution by dividing the squared distance to this directrix by the overall Euclidean distance, that is, \( \frac{d(u)^2}{\sqrt{d(u)^2 + d(v)^2}} \). This directrix of a hyperbola inspired us to design a new geometric topological index, which we introduce here as the Hyperbolic Directrix Sombor Index, its definition is given by: \( \mathcal{HDSO}(G)=\sum_{uv\in E(G)}\frac{\{\min(d(u),d(v))\}^{2}}{\sqrt{d^{2}(u)+d^{2}(v)}} \). We give basic properties of the \( \mathcal{HDSO} \) index and obtain \( \mathcal{HDSO} \) index for few fundamental graphs. We also discuss propeties of octane isomers by relating it to the HDSO index. We propose several structural variants of the \( \mathcal{HDSO} \) index: a modified and a reduced version.