As a Junior Navigation Specialist for a space agency, you are verifying the orbital parameters of Pluto to update a solar system model. The model places the center of the orbit at the origin, with vertices at (40, 0) and $(-40, 0)$, and the Sun at a focus at (10, 0). Match each variable used in the orbital equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with its correct numerical value or definition based on this specific model.

Imagine you are an intern for a space exploration agency documenting the mathematical models of our solar system. Arrange the following steps in the correct order to derive the standard equation of Pluto's elliptical orbit, based on the specific measurements and process provided in your training guide.

You are a junior navigation specialist at a space agency verifying the mathematical models for the solar system. In a specific model of Pluto's elliptical orbit centered at the origin, the vertices are located at $(-40, 0)$ and (40, 0) in astronomical units (AU), and the Sun is positioned at a focus point of (10, 0). According to the standard orbital equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, what is the numerical value for $b^2$?

As a junior orbital mechanics assistant for a space agency, you are verifying the parameters of Pluto's elliptical orbit. True or False: In the standard model where the orbit is centered at the origin (0, 0), the Sun is positioned at the center of the ellipse.

Verifying Orbital Parameters in a Database

As a quality assurance tester for an educational astronomy app, you are verifying the hardcoded mathematical models. In the provided model for Pluto's elliptical orbit centered at the origin, with the Sun at a focus of (10, 0) and a semi-major axis of $a = 40$, the specific numerical value for $b^2$ (which serves as the denominator for the $y^2$ term) is ____.

Documenting Pluto's Orbital Equation for a Science Museum Exhibit