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Analytic Geometry

Analytic geometry is a field of geometry which is represented through the use of coordinates which illustrate the relatedness between an algebraic equation and a geometric structure. In both algebra and geometry, the techniques of analytic geometry are used to solve problems.

Circles, lines and points are the most basal geometric structures which are modelled using analytic geometry. Ordered pairs, such as (x,y) are used to represent points and sets of points, such as (x1,y1) and (x2,y2) are used for describing that a line is related to a linear equation. These points represent the coordinates of where this line passes through. In its basic form, a line in two dimensional space is denoted as ax + by + c = 0.

Furthermore, when dealing with three dimensional spaces, a point consists of three values, (x,y,z). Thus, the linear equation representing a set of coordinates in 3-D space is represented as ax + by + cz + d = 0. This linear equation represents a plane and not a line.

This discussion presents a very brief description on the framework surrounding analytic geometry. Furthermore, it is important to note that analytic geometry has interesting applications in the real-world. For instance, a modern day example of analytic geometry would be how algebraic equations are inputted into computers and manipulated in order to produce geometric structures in the form of animations on a screen. 

Categories within Analytic Geometry

Vector Calculus

Postings: 370

Vector calculus is a field of mathematics which is depicted most commonly in three dimensional spaces and involves utilizing both the operations of differentiation and integration.

Trigonometry

Postings: 845

Trigonometry is a field of geometry which focuses solely on measuring triangles in terms of the lengths of their sides and their angles.

Listing Intercepts and Testing for Symmetry

Please help on how to approach the following problems: 1. For the given equation, list the intercepts and test for symmetry x^2+4y^2=4 What are the intercepts? 2. For the given equation, list the intercepts and test for symmetry. y=negative7x/x^2+25 What are the intercepts? 3. If (a, negative 6) is a point on th

Conic Sections: Analytic Geometric Review

There are 38 attached problems that look like these: Find the midpoint of (5, 0) and (-4, 2). Write the standard form of -2x² + 16x +24y- 224=0 Find the foci of 45y² - 320x²+ 6= 2886

Truth Table Construction

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Fallacy of the Converse

Identify which argument is invalid. If I sing in the shower, then I will not be overheard while singing. I sang in the shower. Therefore, I was not overheard while singing. Either I sing in the shower or I will be overheard while singing. I did not sing in the shower. Therefore, I was overheard while sin

Is [0,1] Closed at the Origin?

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The Best Places to Put the Two Support Poles

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Analytic Functions

Let f:from C to C be analytic. Define g:from C to C by g(x)= ~(f(~z))^2. Show that g is analytic. (Note: "~" here represents an over-bar., i.e., one over the whole set of parentheses and the other just over letter "z" ).

Properties of Relations and an Euler Walk

1. Determine which of the reflexive, symmetric, and transitive properties are satisfied by the given relation R defined over set S. See Appendix A for the definition of reflexive, symmetric, and transitive properties. S={1,2,3} and R={(1,1), (1,2), (2,1), (2,2)} Appendix A Definition A relation R on a set S may have any of t

Matlab : Euler Formula and Runge-Kutta Fourth Order

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Synthetic, Analytic, and Vector Techniques in EUCLIDEAN GEOMETRY

Describe the advantages and disadvantages of the synthetic, analytic, and vector techniques for proving a given theorem in Euclidean Geometry. Give an example of a theorem in Euclidean Geometry that can be proven using synthetic, analytic, and vector techniques. Discuss the implications of instruction using the three techni

Contractible spaces

(See attached file for full problem description with all symbols) --- Let X be a contractible space: a) Show that X is path connected b) Show that any two continuous maps where Y is any topological space, are homotopic. c) Let and be the map defined by . Show that and are homotopic. ---

Analytic Functions; Harmonic Functions; Laplace's Equation

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Euler Path Problem : The Seven Bridges of Konigsberg

In Konigsberg, Germany, a river ran through the city such that in its center was an island, and after passing the island, the river broke into two parts. Seven bridges were built so that the people of the city could get from one part to another. A crude map of the center of Konigsberg might look like this: The people wondere

Analytical Geometry - Find the locus of a vertex.

The equation of the circle is given as: x^2+y^2=25 A parallelogram is constructed as follows: Vertex O is at the origin, vertex A is on the circle, vertex C is on the y-axis and the diagonal AC is parallel to the x-axis. See attached figure Question: 1. Find the locus of vertex B. 2. Describe the geometrical shap

Find the vertex focus, and directrix of the parabola

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