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Ellipsoid geometry is all about smooth, symmetrical surfaces. Imagine an inflated balloon or a stretched sphere — that's what an ellipsoid looks like. It's a three-dimensional shape with three axes of different lengths. Each axis represents a direction: x, y, or z, and they intersect at the center point of the ellipsoid. This center point is like the heart of the ellipsoid. For simplicity, we often place this point at the origin in coordinate space, which is the center of our geometric universe.

An ellipsoid's equation captures its symmetry and size. The longer the axis, the more elongated the ellipsoid is in that direction. Ellipsoids come up in many fields, including physics and engineering. Understanding them is a core skill in geometry because they help us model real-world objects. For example, Earth's shape can be modeled as an ellipsoid rather than a perfect sphere because of its slight bulging at the equator.

Coordinates for an ellipsoid are based on its equation, which is written as \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \). In this equation, \(a\), \(b\), and \(c\) correspond to the semi-axes, which are half the lengths of the whole ellipsoid in the x, y, and z dimensions respectively.

When we say an ellipsoid is centered at the origin, we mean its central point or center of symmetry is at (0,0,0). This makes it easier to reference different points on the ellipsoid. To describe any point on its surface, you can plug its coordinates into the ellipsoid's equation. Points like A(2,0,0), B(0,0,1), and C\( \left(\frac{1}{2}, \sqrt{11}, \frac{1}{2}\right) \) each meet the required property: when substituted into this equation, they satisfy it completely.

The use of coordinates allows us to verify points on the shape, ensuring accuracy when plotting or modeling ellipsoidal objects.

Mathematical substitution is a vital tool for solving equations in geometry. It involves replacing variables in an equation with specific values or expressions to simplify and solve it. In our case, to find the equation of the ellipsoid passing through given points, we begin by substituting these points into the standard ellipsoid equation.

For instance, with point A at (2,0,0), substitution into \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \) gives \( \frac{4}{a^2} = 1 \). This shows that \( a = 2 \). Similarly, substituting B(0,0,1) gives \( \frac{1}{c^2} = 1 \), indicating \( c = 1 \).

We repeat this process for all given points to calculate each semi-axis. This step is crucial because it directly influences the ellipsoid's shape and spatial orientation. Substitution helps break down complex determinations into manageable computations, ensuring clarity.