Question

Name and sketch the graph of each of the following equations in three-space. $$ 9 x^{2}+25 y^{2}+9 z^{2}=225 $$

Short answer

An ellipsoid centered at the origin, elongated along the x and z-axis.

Step-by-step solution

Recognize the Equation Form

This is an equation of an ellipsoid. The general form of an ellipsoid centered at the origin in three-space is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\).

Rewrite the equation in standard form

Divide the entire equation by 225 to get it in standard ellipsoid form: \( \frac{x^2}{25} + \frac{y^2}{9} + \frac{z^2}{25} = 1 \).

Identify the semi-axes lengths

The values \(a^2 = 25\), \(b^2 = 9\), \(c^2 = 25\) lead to semi-axis lengths of \(a = 5\), \(b = 3\), \(c = 5\).

Analyze the shape and orientation

This ellipsoid is elongated along the x and z-axis (each with semi-axis length 5) and is shorter along the y-axis (semi-axis length 3).

Sketch the Ellipsoid

Sketch an ellipsoid centered at the origin (0,0,0), stretched wider along the x-axis and z-axis each with a radius of 5, and narrower along the y-axis with a radius of 3.

Key concepts

Three-Dimensional Graphs

Three-dimensional graphs are a way to visually represent equations that involve three variables, typically corresponding to the x, y, and z axes in space. Unlike two-dimensional graphs, which lay flat on a plane, three-dimensional graphs extend into the depth, giving them a whole new dimension. This extension allows for more complex shapes and interactions between variables, such as surfaces and solids.

When plotting a three-dimensional graph, it’s important to understand the orientation of each axis. Typically:

• The x-axis runs horizontally.
• The y-axis runs vertically.
• The z-axis extends outward or inward, representing depth.

To graph complex shapes like an ellipsoid, we depict how the equation fills space. For instance, in the context of our ellipsoid equation, each point \(x, y, z\) satisfying the equation lies on the surface of an ellipsoid. This visualization helps understand the orientation and symmetry of multi-variable equations in three-space.

Analytic Geometry

Analytic geometry, also known as coordinate geometry, uses algebraic equations to describe geometric shapes. This branch of mathematics bridges algebra and geometry, allowing us to solve geometric problems using equations and vice versa.

In analytic geometry, points are identified using coordinates. For three-dimensional problems, each point in space is described using three coordinates: \(x, y, z\). These coordinates are used to define shapes, lines, surfaces, and volumes mathematically.

The equation \(9x^2 + 25y^2 + 9z^2 = 225\) represents an ellipsoid in three-dimensional space. By manipulating this equation, such as dividing through by 225, we express it in a standard form that highlights the relations between x, y, and z. This standard form is crucial in analytic geometry as it simplifies the complexity of shapes, making it easier to understand their properties and relationships.

Conic Sections

Conic sections arise from slicing a cone in different ways. They include familiar shapes like circles, ellipses, parabolas, and hyperbolas. While conic sections are typically discussed in two dimensions, extending these concepts into three dimensions leads to interesting shapes, like ellipsoids.

An ellipsoid is a three-dimensional analogue of an ellipse. In standard form, an ellipsoid's equation resembles that of an ellipse, involving three squared terms, each divided by a constant. This form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\) characterizes the ellipsoid and determines its size and shape. In the context of the given exercise, the ellipsoid is recognized by noting the squared terms and understanding how they define the axes' lengths.

By comparing the original equation to this form, we identify the lengths of the semi-axes, allowing us to visualize how the ellipsoid is stretched over three dimensions. Understanding these relationships is key when exploring three-dimensional conic sections because they highlight the symmetry and dimensions of the shape.