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Conic sections are curves obtained by intersecting a cone with a plane. These sections form distinct shapes such as circles, ellipses, parabolas, and hyperbolas. An ellipsoid, which is a three-dimensional extension, is closely related to an ellipse. Similar to how a circle is to an ellipse, a sphere is to an ellipsoid. When dealing with an ellipsoid, think of it as a stretched sphere where each stretch is determined by its semi-axis lengths in the x, y, and z directions. This concept is fundamental in understanding the geometry of 3D shapes.

The standard form of an ellipsoid is crucial for correctly identifying its parameters. For an ellipsoid centered at the origin, the equation is given by \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \]This equation tells us how the space is divided along the three primary axes. The values under each squared term, \(a^2, b^2,\) and \(c^2\), represent the squares of the semi-axis lengths. The numbers themselves indicate how far the ellipsoid extends in each direction from the origin. By substituting points known to lie on the ellipsoid into this equation, we can determine these semi-axis lengths.

Semi-axis lengths are vital components that define the size and shape of an ellipsoid. When examining the standard form equation, each term's denominator represents a squared semi-axis length:

• \(a\) is the semi-axis length along the x-axis.
• \(b\) is along the y-axis.
• \(c\) is along the z-axis.

In this exercise, substituting the points into the standard form equation gives us \(a^2 = 4\), \(b^2 = 16\), and \(c^2 = 1\). The square roots of these values (\(a\), \(b\), \(c\)) give the actual semi-axis lengths, indicating how the ellipsoid stretches along each axis. It's important to understand that these lengths determine the ellipsoid's physical dimensions and orientation.

Coordinate geometry, or analytic geometry, involves using algebra to solve geometric problems. In the context of ellipsoids, it helps us visualize and calculate their properties based on given points. By employing coordinate geometry, we can manipulate the ellipsoid's standard form equation to unravel the geometric properties such as its semi-axis lengths.
In this exercise, we effectively put coordinate geometry to use by manipulating the coordinates of points \(A\), \(B\), and \(C\) to find the equation of the ellipsoid. This application highlights the power of coordinate geometry in transitioning from numeric representations to visual, spatial understanding鈥攁 core principle in both mathematics and its practical applications. It seamlessly bridges abstract algebra with tangible geometry insights.