Chapter 19: Problem 4
The relation \(|3-x|+|3+z|=5\) represents (a) a circle (b) a parabela (c) an ellipse (d) a hyperbola.
Short Answer
Expert verified
None of the above; it's a rhombus-like shape, not a conic section.
Step by step solution
01
Understanding Absolute Value
The expression involves two absolute values: \(|3-x|\) and \(|3+z|\). Recall that the absolute value \(|a|\) represents the distance of \(a\) from zero on the number line, meaning \(|a|\geq 0\) for all \(a\).
02
Analyzing Range of Values
Each absolute value function is non-negative, so together they can range from \(0\) to \(5\). This means the physical interpretation is the set of points that can combine distances to make a total distance of \(5\).
03
Solving for Geometry
The given equation \(|3-x| + |3+z| = 5\) is akin to the equation of a diamond or lozenge shape which is described in a plane as a rhombus or a dilated version of a rotated square. Explicitly, it represents the various ways to split distance \(5\) between two different axes, resembling a diamond on the coordinate plane without curving nature inherent in circles, ellipses, or parabolas.
04
Identifying the Conic
This equation does not match the standard forms of conic sections like ellipses, parabolas, hyperbolas, or a circle. Rather, it's the geometric locus of points with a fixed sum of distances to two lines (specifically the x-axis shifted by 3). This does not naturally fit into any of the types listed as it forms a different kind of locus.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Absolute Value Equations
Absolute value equations like \(|3-x| + |3+z| = 5\) relate to distances. In mathematics, the absolute value function, denoted as \(|a|\), gives the distance of a number from zero. This distance is always a non-negative figure, meaning \(|a| \geq 0\) for every real number \(a\).
These types of equations can define a set of points in a plane. Specifically, they describe scenarios where the total combined distance from paired numbers equals a specific value - here, it's 5, dissecting the total sum between separate sections. In simpler terms, consider the absolute values as measuring sticks attached horizontally and vertically, outstretching to maintain a constant total length regardless of direction.
Absolute value equations are fundamental tools in geometry. They help in, among other things, plotting geometric shapes without dependent curvatures, especially those that form linear (straight line) confines or boundaries. This equation sets the stage to explore the essence of purity in mathematical distances.
These types of equations can define a set of points in a plane. Specifically, they describe scenarios where the total combined distance from paired numbers equals a specific value - here, it's 5, dissecting the total sum between separate sections. In simpler terms, consider the absolute values as measuring sticks attached horizontally and vertically, outstretching to maintain a constant total length regardless of direction.
Absolute value equations are fundamental tools in geometry. They help in, among other things, plotting geometric shapes without dependent curvatures, especially those that form linear (straight line) confines or boundaries. This equation sets the stage to explore the essence of purity in mathematical distances.
Geometric Loci
Geometric loci are fascinating as they describe all points satisfying certain conditions on a plane. Often, they are related to geometric shapes we encounter frequently in conic sections like circles, ellipses, parabolas, and hyperbolas.
In the case of \(|3-x| + |3+z| = 5\), the locus relates to two absolute value expressions adding up to a constant sum. This effectively implies that every solution to the equation is a point located such that its algebraic distance meets the preset sum condition. Imagine walking along a path, constantly checking a string stretched to balance at exactly 5 units for your position. Wherever this balance happens meets the loci condition.
These geometric ideas extend deeper meanings into obscure shapes, revealing possibilities even outside standard conic representations. It teaches us the broader concept of geometric placement beyond known shapes, offering a holistic view of mathematical space.
In the case of \(|3-x| + |3+z| = 5\), the locus relates to two absolute value expressions adding up to a constant sum. This effectively implies that every solution to the equation is a point located such that its algebraic distance meets the preset sum condition. Imagine walking along a path, constantly checking a string stretched to balance at exactly 5 units for your position. Wherever this balance happens meets the loci condition.
These geometric ideas extend deeper meanings into obscure shapes, revealing possibilities even outside standard conic representations. It teaches us the broader concept of geometric placement beyond known shapes, offering a holistic view of mathematical space.
Non-Standard Conic Sections
Non-standard conic sections represent forms not captured by typical conics like circles, ellipses, parabolas, or hyperbolas. Equations that don't comfortingly fall into conventional categories often describe unique geometric properties that enrich our understanding.
The equation \(|3-x| + |3+z| = 5\) transcends these standard definitions. Instead, it shapes an unconventional locus—a diamond or lozenge, captured from how absolute values assign linear paths confined within calculated boundaries. It symbolizes how mathematical equations expand the concept of geometric designs beyond what might customarily be imposed.
Such explorations illustrate diverse mathematical landscapes, where solving and understanding such equations contribute to recognizing and appreciating abstract geometrical forms. Non-standard conic sections help cultivate a broader view of mathematical applications and encourage critical thought regarding unseen possibilities.
The equation \(|3-x| + |3+z| = 5\) transcends these standard definitions. Instead, it shapes an unconventional locus—a diamond or lozenge, captured from how absolute values assign linear paths confined within calculated boundaries. It symbolizes how mathematical equations expand the concept of geometric designs beyond what might customarily be imposed.
Such explorations illustrate diverse mathematical landscapes, where solving and understanding such equations contribute to recognizing and appreciating abstract geometrical forms. Non-standard conic sections help cultivate a broader view of mathematical applications and encourage critical thought regarding unseen possibilities.
