The normals to the ellipse \(\displaystyle \frac{x^2}{4} + \frac{y^2}{9} =1 \) at \(\displaystyle (x_k , y_k ) ; k = 1,2,3,4 \) are concurrent.

Evaluate: \(\displaystyle \sum_{m=1}^4 \sum_{n=1}^4 \frac{x_m }{x_n } \)

**Details and Assumptions:**

The points \( (x_k , y_k) ; k =1,2,3,4 \) all lie on the given ellipse.

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