Problem 32 - APEX Calculus

Question

Sketch the quadric surface. \(4 x^{2}+2 y^{2}+z^{2}=4\)

Step-by-Step Solution

Answer

The surface is an ellipsoid centered at the origin.

Step 1: Rewrite the Equation in Standard Form

Start by dividing all terms of the equation \(4x^2 + 2y^2 + z^2 = 4\) by 4 to simplify it. This gives: \(\frac{x^2}{1} + \frac{y^2}{2} + \frac{z^2}{4} = 1\). This is the standard form of a quadric surface.

Step 2: Identify the Type of Quadric Surface

The equation \(\frac{x^2}{1} + \frac{y^2}{2} + \frac{z^2}{4} = 1\) matches the standard form of an ellipsoid, which is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\). Thus, the surface is an ellipsoid with semi-axis lengths \(a = 1\), \(b = \sqrt{2}\), and \(c = 2\).

Step 3: Sketch the Ellipsoid

1. Draw the coordinate axes: x, y, and z.2. Plot points at the ends of each semi-axis: (1, 0, 0), (-1, 0, 0) for x-axis, (0, \(\sqrt{2}\), 0), (0, -\(\sqrt{2}\), 0) for y-axis, and (0, 0, 2), (0, 0, -2) for z-axis.3. Connect these points with an oval shape representing the ellipsoid.The sketch should be symmetrical about each of the axes, reflecting how lengths are scaled by the values of a, b, and c.

Key Concepts

Ellipsoid

An ellipsoid is a type of quadric surface that is a three-dimensional shape. Imagine a roundish stretched-out sphere. Much like how a circle turns into an ellipse when drawn flat, a sphere becomes an ellipsoid when extended in different directions. The defining characteristic of an ellipsoid is its symmetry about three perpendicular axes. This symmetry is deeply rooted in the equation used to describe it.
For an ellipsoid, the general equation in standard form is given by:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \)

Here, \(a\), \(b\), and \(c\) represent the lengths of the semi-axes along the x, y, and z dimensions, respectively. The given exercise illustrates this with an equation that, once simplified, reveals its nature as an ellipsoid. Recognizing an ellipsoid in mathematical problems involves understanding how these axes differ from those of a perfect sphere.

Standard Form

Mathematical equations often have a 'standard form', which simplifies analysis and identification. For an ellipsoid, the standard form helps in easily spotting the type of surface and determining its key properties. In the given problem, the starting equation is \(4x^2 + 2y^2 + z^2 = 4\). By dividing all terms by the number on the right-hand side, 4, it transforms into:

• \( \frac{x^2}{1} + \frac{y^2}{2} + \frac{z^2}{4} = 1 \)

This transformation puts the equation in standard form, making it apparent that it corresponds to an ellipsoid. The standard form helps in clear identification because it aligns with the generic equation of quadric surfaces. This process of converting to standard form is key in identifying the surface type—something students should practice to make it second nature.

Semi-Axis Lengths

The semi-axis lengths \(a\), \(b\), and \(c\) are fundamental to defining the shape of an ellipsoid. They represent how far the ellipsoid stretches along each axis from the center to the surface. These lengths tell us the dimensions and proportions of the shape.

• For the x-axis, we have \(a = \sqrt{1} \), which simplifies to 1.
• On the y-axis, \(b = \sqrt{2} \) shows an intermediate length.
• The z-axis is the longest with \(c = \sqrt{4} = 2 \).

The difference in these lengths shows how the ellipsoid is elongated in specific directions. Imagine plotting points along each axis at ± the semi-axis lengths and connecting these points, forming an oval shape. This visualization helps in sketching the ellipsoid accurately. Understanding semi-axis lengths is crucial for visualizing and working with ellipsoids in problems.