Question

Give a geometric description of the following sets of points. $$x^{2}-4 x+y^{2}+6 y+z^{2}+14=0$$

Short answer

1. Rewrite the equation in standard form by completing the squares. 2. Identify the type of surface based on the coefficients and exponents. 3. Conclude that the given equation does not represent any real geometric shape in Euclidean space since the right-hand side is negative.

Step-by-step solution

Rewrite the equation in standard form

To better see the type of surface, rewrite the given equation such that each term has a common factor. In this case, we will complete the squares for the x, y, and z components to get closer to a standard form for a quadric surface. The given equation is: $$x^{2}-4 x+y^{2}+6 y+z^{2}+14=0$$ For our x terms, add and subtract \((4/2)^2=4\): $$x^2 - 4x + 4 = (x-2)^2$$ For our y terms, add and subtract \((6/2)^2=9\): $$y^2 + 6y + 9 = (y+3)^2$$ Now, rewrite the equation by plugging the completed squares and moving the constants to the other side of the equation: $$ (x-2)^2 + (y+3)^2 + z^2 = 4 + 9 - 14$$ $$ (x-2)^2 + (y+3)^2 + z^2 = -1 $$

Identify the type of surface

Now that we have our equation in the form $$ (x-2)^2 + (y+3)^2 + z^2 = -1$$ We can see that all variables have positive coefficients with equal exponents (2). We can identify this as an equation of a sphere; however, since the right-hand side is negative, this is impossible since the sum of squared variables can never be negative. Therefore, our conclusion is this equation does not represent any real geometric shape in Euclidean space.

Key concepts

Completing the Square

Understanding how to manipulate quadratic equations is essential in calculus and algebra. One common method is 'completing the square', which helps to transform a quadratic equation into a simpler form that can reveal the nature of the graph representing the equation.

Completing the square involves creating a perfect square trinomial from a quadratic expression, thus making it easier to solve or analyze. In the given exercise, the student must complete the square for both the x and y variables. Here's how it works:

• Consider the equation: \( ax^2 + bx + c \).
• To complete the square, you find \( (b/2)^2 \) and add and subtract it within the equation.
• This forms a perfect square trinomial \( (x + b/2)^2 \), which can then be used to rewrite the original equation.

The objective is to reorganize the equation to resemble a well-known form, often leading to the identification of the graph's shape, such as a circle, ellipse, or parabola.

In our example, for the x-term, \( (4/2)^2 = 4 \) is added and subtracted yielding \( (x-2)^2 \), and similarly for y, \( (6/2)^2 = 9 \) to get \( (y+3)^2 \). The goal of this approach is to reveal the underlying structure of the quadric surface represented by the equation.

Geometric Description of Equations

Equations in calculus often correspond to geometric shapes or surfaces. When we have an equation like \( x^2 + y^2 + z^2 + \text{ ...} = 0 \), our job is to interpret this as a geometric figure. This process is known as giving a 'geometric description' of an equation.

For a proper analysis, it's crucial to manipulate the equation into a standard form, which often requires techniques like completing the square. Once in standard form, we can compare it to the equations of known shapes and determine the graph's nature, such as whether it's a plane, sphere, cylinder, etc.

Unfortunately, in the exercise provided, after completing the square and rearranging the equation, we ended up with \( (x-2)^2 + (y+3)^2 + z^2 = -1 \), which suggests a nonexistent shape in real Euclidean geometry since there are no real number solutions for a set of squared numbers that add up to a negative number. This underlines the importance of both algebraic manipulation and an understanding of geometric constraints.

Types of Surfaces in Calculus

In calculus, understanding the types of surfaces represented by equations is crucial for visualizing and solving three-dimensional problems. Common types of surfaces include:

• Planes: Simplest surfaces defined by linear equations.
• Spheres: All points equidistant from a central point, given by \( (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2 \) where \((h,k,l)\) is the center and \(r\) is the radius.
• Ellipsoids, paraboloids, and hyperboloids: Defined by quadratic equations, these surfaces can be found by setting, modifying, and analyzing the sum or difference of squared variables.
• Cylinders: Surfaces with a consistent cross-section along a particular axis; can be elliptic, hyperbolic, or parabolic.
• Cones: Defined by a linear relationship between squared variables and can be seen in equations like \( z^2 = ax^2 + by^2 \).

These surfaces are known as 'quadric' surfaces because they can usually be described by second degree polynomial equations. In the implicit exercise, we were initially led to believe the equation might represent a sphere. However, after examination and completing the square, we found that no real solutions exist for the given equation, hence it doesn't represent a real quadric surface.