Let $P=(X_1,\dots,X_p)$ and $Q=(Y_1,\dots,Y_p)$ two vectors of random variables. The statistical distance between $P$ and $Q$ is $$D(P,Q)=\sqrt{a_{11}(x_1-y_1)^2+\dots+a_{pp}(x_p-y_p)^2+2a_{12}(x_1-y_1)(x_2-y_2)+\dots+2a_{p-1p}(x_{p-1}-y_{p-1})(x_p-y_p)}$$
where the $a'$s are numbers such that the distances are always nonnegative and $D^2(P,Q)$ is a elipse centered in $Q$
From what I read for $p=2$ the $a's$ are $a_{11}=\frac{1}{s_{11}}$, $a_{22}=\frac{1}{s_{22}}$ and $a_{12}=\frac{1}{s_{12}}$, where $s_{11}=\frac{1}{n}\sum (x_{i1}-\overline{x_1})^2$ is the sample variance and $s_{12}=\frac{1}{n}\sum (x_{i1}-\overline{x_1})(x_{i2}-\overline{x_2})$ is the sample covariance.
So I think that for $p$ random variables $a_{ij}=\frac{1}{s_{ij}}$. I know that this measures the distance between two vectors of random variables, but I can't understand why it have this form. What is the idea behind it? Why $D^2(P,Q)$ defines a elipse centered in $Q$.