Question

Give a geometric description of the following sets of points. $$x^{2}-2 x+y^{2}+6 y+z^{2}+10=0$$

Short answer

The given equation represents an ellipsoid with its center at the point (1, -3, 0) and semi-axes of length √2 along the x, y, and z axes.

Step-by-step solution

Rewrite the equation

Rearrange the equation to make it easier to complete the square: $$x^{2} - 2x + y^{2} + 6y + z^2 + 10 = 0$$

Complete the square for x and y

To do this, we need to add and subtract the necessary terms to the equation: $$\left(x^{2} - 2x + 1\right) + \left(y^{2} + 6y + 9\right) + z^2 = 0 + 1 - 9 + 10$$ The rewritten equation is now: $$(x - 1)^{2} + (y + 3)^{2} + z^2 = 2$$

Analyze the resulting equation

We have an equation of the form: $$(x - a)^2 + (y - b)^2 + (z - c)^2 = d^2$$ This is the equation for an ellipsoid centered at the point (a, b, c) with semi-axes along the x, y, and z axes, of lengths equal to the square root of the term on the right side (d). In our case, the ellipsoid will have its center at (1, -3, 0) and semi-axes of lengths \(\sqrt{2}\) along each axis.

Write the geometric description

The given set of points represents an ellipsoid centered at (1, -3, 0) with semi-axes of length \(\sqrt{2}\) along the x, y, and z axes.

Key concepts

Completing the Square

Completing the square is a useful technique in algebra for transforming quadratic expressions into a specific form that makes them easier to work with. This process involves rewriting a quadratic expression in terms of a perfect square trinomial plus or minus a constant. Let's see how this technique can simplify our equation:

• Take the expression involving a variable, like \(x^2 - 2x\). To complete the square, identify the coefficient of the linear term, which is \(-2\) in this case, and take half of it, which is \(-1\), and then square it. This results in \(1\).
• Add and subtract \(1\) inside the equation: \((x^2 - 2x + 1 - 1)\).
• This allows us to rewrite \(x^2 - 2x + 1\) as \((x - 1)^2\).

The same procedure is applied to \(y^2 + 6y\). Completing the square helps to transform the original equation into a recognizable form, which is essential for understanding its geometric interpretation.

Geometric Description

The equation derived from completing the square describes a specific 3D shape. Here, the resulting expression \((x - 1)^2 + (y + 3)^2 + z^2 = 2\) corresponds to an ellipsoid in 3D space. An ellipsoid is similar to an elongated sphere. It is a closed surface where all cross-sections are ellipses or circles.

• In this case, the center of the ellipsoid is at the point \((1, -3, 0)\), determined by the values \(a = 1\), \(b = -3\), and \(c = 0\) from \((x - a)^2 + (y - b)^2 + (z - c)^2\).
• The right side of the equation provides the radius squared, which is \(2\), leading to a radius \(\sqrt{2}\). This radius applies equally to each axis, indicating the axes have equal lengths.

Understanding the geometric description helps visualize how a mathematical expression translates into a shape within 3D space.

3D Coordinates

In mathematics, 3D coordinates allow us to pinpoint a location in three-dimensional space. Each point in 3D space is defined by an ordered triplet \((x, y, z)\). These coordinates correspond to the position along the x, y, and z axes, respectively.

• For our transformed equation, the center of the ellipsoid is given by the coordinates \((1, -3, 0)\). This center is crucial for constructing and understanding the positioning of the ellipsoid.
• An important part of working with 3D coordinates is visualizing how shifts in these coordinates affect the shape and location of geometric figures.
• Here, each shift along the x, y, or z-coordinate helps to understand how the ellipsoid is oriented and how it extends in 3D space.

Using 3D coordinates, we can map and graph the position of geometric objects accurately, offering deeper insights into their properties and behaviors.