Circle / Ellipse Hybrid

Consider a circle and an ellipse:

$\large{x^2 + y^2 = 1 \\ \frac{x^2}{4} + \frac{y^2}{1} = 1}$

A point on either shape could be represented in polar coordinates, with $r_C (\theta)$ and $r_E (\theta)$ being the respective radii of the circle and ellipse as a functions of $\theta$ . Let $A_C$ and $A_E$ be the areas of the circle and ellipse, respectively.

Suppose we make a hybrid shape, defined as follows:

$\large{r ( \theta ) = \sigma \, r_C (\theta) + (1- \sigma) \, r_E (\theta) \\ A = \frac{A_C + A_E}{2}}$

If $r (\theta)$ is the radius of the hybrid shape, $\sigma$ is a constant, and $A$ is the hybrid shape area, what is $\lfloor 1000 \, \sigma \rfloor$ ?

Note: $\lfloor \cdot \rfloor$ denotes the floor function