A satellite technician is modeling an elliptical orbit with the equation $\frac{x^2}{625} + \frac{y^2}{600} = 1$, where the center is at the origin and the major axis is horizontal. In this standard form, what geometric property of the orbit is represented by the denominator 625?

A mission specialist at a satellite tracking station is modeling an elliptical orbit centered at the origin $(0, 0)$. In this specific model, the vertices are located at $(\pm 25, 0)$ and the sun is positioned at a focus point $(5, 0)$. Match each orbital parameter with its corresponding value or formula used to determine the standard equation of the orbit.

Orbital Equation Parameters

An astrophysics researcher is documenting the procedure for modeling a planet's elliptical orbit where the sun is at a focus, with a closest distance of 20 AU and a furthest distance of 30 AU. Arrange the following steps in the correct order to derive the standard equation of the ellipse centered at the origin, as demonstrated in the lab manual.

A data analyst at a satellite tracking station is modeling a planet's elliptical orbit centered at the origin. The analyst identifies that the squared semi-major axis is $a^2 = 625$ and the squared distance to the focus is $c^2 = 25$. Using the relationship $b^2 = a^2 - c^2$ to find the denominator for the $y^2$ term, the analyst determines that the value of $b^2$ is ____.