Problem 38
Question
Sketch the quadric surface. \(\frac{x^{2}}{4}+y^{2}+\frac{z^{2}}{9}=1\)
Step-by-Step Solution
Verified Answer
The quadric surface is an ellipsoid with axes lengths 2, 1, and 3.
1Step 1: Identify the Type of Quadric Surface
The given equation is \(\frac{x^{2}}{4}+y^{2}+\frac{z^{2}}{9}=1\). This equation is in the form of \(\frac{x^{2}}{a^2} + \frac{y^{2}}{b^2} + \frac{z^{2}}{c^2} = 1\), which represents an ellipsoid. The denominators \(a^2 = 4\), \(b^2 = 1\), and \(c^2 = 9\) show the semi-axis lengths.
2Step 2: Determine the Axes Lengths
From the equation, compare each squared term to its respective axis. The semi-axis length for \(x\) is \(a=2\), for \(y\) is \(b=1\), and for \(z\) is \(c=3\). These values represent how wide the ellipsoid stretches along each axis.
3Step 3: Sketch the Ellipsoid
To sketch the ellipsoid, plot the points along the x, y, and z axes at distances corresponding to their semi-axes lengths: \(\pm 2\) for \(x\), \(\pm 1\) for \(y\), and \(\pm 3\) for \(z\). Connect these boundary points smoothly to visualize the shape of an ellipsoid, which is symmetric about the origin.
Key Concepts
EllipsoidEquation of Quadric SurfacesAxes Lengths
Ellipsoid
An ellipsoid is a three-dimensional shape that's a natural extension of an ellipse. Similar to how an ellipse looks like a stretched circle, an ellipsoid appears like a stretched sphere. Ellipsoids are characterized by their symmetry around three perpendicular axes: the x, y, and z axes. They are smooth, convex surfaces where every cross-section is an ellipse or a circle.
To understand ellipsoids, you can imagine starting with a sphere and stretching it by different amounts along each axis. The lengths by which an ellipsoid is elongated along these three axes define its shape. This makes ellipsoids incredibly versatile in various disciplines, from geometry to physics, as they model how objects might stretch in space.
Ellipsoids can come in various forms depending on the relative lengths of these axes, but they always exhibit a continuous and smooth surface. When visualizing an ellipsoid, you often see it plotted symmetrically around the origin of the axes.
To understand ellipsoids, you can imagine starting with a sphere and stretching it by different amounts along each axis. The lengths by which an ellipsoid is elongated along these three axes define its shape. This makes ellipsoids incredibly versatile in various disciplines, from geometry to physics, as they model how objects might stretch in space.
Ellipsoids can come in various forms depending on the relative lengths of these axes, but they always exhibit a continuous and smooth surface. When visualizing an ellipsoid, you often see it plotted symmetrically around the origin of the axes.
Equation of Quadric Surfaces
The equation of quadric surfaces is fundamental in describing different geometric surfaces in three dimensions. These surfaces include spheres, cylinders, hyperboloids, and importantly, ellipsoids. The general form of a quadric surface equation is expressed as:
For specific cases, like an ellipsoid, the equation simplifies to a form that enables easier calculation and visualization. This form is
By reducing the full quadric expression down to these primary terms, we can isolate the key components that define the surface's shape, making it practical to handle and sketch in exercises. This equation captures the proportional dimensions of the ellipsoid, dictating how much it stretches along each axis.
- \[ Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0 \]
- Here, each term represents the quadrature (squared terms) mixed with linear effects to form complex surfaces.
For specific cases, like an ellipsoid, the equation simplifies to a form that enables easier calculation and visualization. This form is
- \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \]
- Each variable \(x, y, z\) is divided by its corresponding squared axis length \(a^2, b^2, c^2\).
By reducing the full quadric expression down to these primary terms, we can isolate the key components that define the surface's shape, making it practical to handle and sketch in exercises. This equation captures the proportional dimensions of the ellipsoid, dictating how much it stretches along each axis.
Axes Lengths
Axes lengths are essential to understanding and defining the shape of an ellipsoid. In the context of the equation of an ellipsoid, the terms \(a, b,\) and \(c\) represent the semi-axis lengths, based on the equation
When you take each denominator under the squared variables, you derive these lengths. For example, if you have an equation part like \(\frac{x^2}{4}\), you identify that \(a^2 = 4\), leading to a length \(a = 2\). This calculation is repeated for each variable, yielding the full set of axis lengths:
These lengths are crucial for sketching or visualizing the ellipsoid, as they give a precise framework from which the figure can be plotted symmetrically around the origin. They provide a straightforward means to visualize the size and proportions of the ellipsoid in three-dimensional space.
- \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \].
- These values translate directly to how far the ellipsoid extends along each axis from the origin.
When you take each denominator under the squared variables, you derive these lengths. For example, if you have an equation part like \(\frac{x^2}{4}\), you identify that \(a^2 = 4\), leading to a length \(a = 2\). This calculation is repeated for each variable, yielding the full set of axis lengths:
- Length along the x-axis: \( a = 2 \)
- Length along the y-axis: \( b = 1 \)
- Length along the z-axis: \( c = 3 \)
These lengths are crucial for sketching or visualizing the ellipsoid, as they give a precise framework from which the figure can be plotted symmetrically around the origin. They provide a straightforward means to visualize the size and proportions of the ellipsoid in three-dimensional space.
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