This problem combines several key concepts of ellipses: identifying its standard form, calculating its foci and eccentricity, understanding the equation of a chord of contact, and applying the focal distance property. Our goal is to find the sum of the squared distances from a specific focus to the points of tangency.

1. Analyze the Ellipse and its Properties

First, let's identify the characteristics of the given ellipse: The equation is $\frac{x^{2}}{2}+\frac{y^{2}}{4}=1$.

• Standard Form: Compare this to the standard form of an ellipse $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ (for a vertical major axis) or