91Ó°ÊÓ

A sphere is a three-dimensional object where every point on the surface is equidistant from a central point, known as the center of the sphere. The constant distance from the center to any point on the sphere is called the radius. The general equation of a sphere centered at a point \((h, k, l)\) with radius \(r\) is:

• \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\)

In the exercise provided, after completing the square, the equation is transformed into a form that looks like:

• \((x - \frac{3}{2})^2 + (y + 2)^2 + (z - 4)^2 = 0\)

This tells us that the sphere is centered at \((\frac{3}{2}, -2, 4)\), and the radius is 0. A radius of zero implies that the sphere is not extended in space but rather reduced to a single point at its center. This is an important concept as it ties into understanding the geometrical properties of spheres in different mathematical contexts.

Completing the square is a technique used to transform quadratic expressions into a perfect square trinomial. This is particularly helpful in simplifying equations and is extensively used in algebra to solve quadratic equations, integrate functions, and describe conic sections.

• The process involves taking a quadratic expression like \(ax^2 + bx + c\) and rearranging it to become \(a(x - h)^2 + k\).
• This involves finding a value \(d\) such that \((x - d)^2\) mirrors the original quadratic expression, which is achieved by manipulating the coefficient of the linear term.

In the original problem, completing the square is done separately for each of the variables \(x\), \(y\), and \(z\). For instance, the term \(x^2 - 3x\) is rewritten as \((x - \frac{3}{2})^2 - \frac{9}{4}\). By completing the square for each variable, the equation of the sphere was found in its standard form. This process helps in recognizing and rewriting the equation to easily identify its geometric representation.

Conic sections are the curves obtained as the intersection of a cone with a plane. Depending on how the plane intersects the cone, several types of conic sections can be formed:

• Circle: A special type of ellipse where its two axes are of equal length.
• Ellipse: Formed when the plane intersects the cone at an angle, resulting in an oval shape.
• Parabola: Produced when a plane is parallel to the slope of the cone.
• Hyperbola: Occurs when the plane intersects both nappes of the cone.

The equation given in the textbook initially seems to resemble that of a conic section due to its quadratic nature. However, through steps of completing the square, it is simplified to a sphere equation. This showcases that while the starting form of an equation might suggest multiple interpretations (as either a conic section or a sphere), further algebraic manipulation clarifies its true nature. Conic sections arise in various scientific fields, and understanding them provides a foundation for interpreting complex shapes and equations.