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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.

## BrainMass Categories within Analytic Geometry

### Vector Calculus

Solutions: 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

Solutions: 846

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 -2xÂ² + 16x +24y- 224=0 Find the foci of 45yÂ² - 320xÂ²+ 6= 2886

### Truth Table Construction

11. Write the statement in symbols using the p and q given below. Then construct a truth table for the symbolic statement and select the best match. p = I eat too much q = I'll exercise. I'll exercise if I eat too much. A. q -> p p q q p T T T T F T F T F F F T

### Equation of a Sphere and Geometric Plane

I have the solution to these problems. Solution is included but need the calculations to find these solutions. Please see attached document.

### 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

### what is the number of subsets of set A

Without writing them all out, what is the number of subsets of set A = {tongue, ear, mouth, eye, nose, cheek, forehead, neck, shoulder}?

### Is [0,1] Closed at the Origin?

Please solve the following problem: Let C ([0,1]) be the space of continuous functions in [0,1] with the norm II f II = max I f(x) I on [0,1]. Is the subspace of functions that are = 0 at the origin closed subspace in C^0 [o,1] with this norm/Prove or disapprove.

### The Best Places to Put the Two Support Poles

Here is the specific question: A landscape architect is drawing plans for a rigid lightweight canopy to provide shade for a patio. Two poles will support the canopy. On the drawing, the coordinates of the verticies of the canopy are: P (0,0) Q ( 2,3) R (8, -1) and S (6, -4) What are the best places to put the two sup

### Euler diagram and argument

Some artists like drawing Some artists like painting _____________________________ Therefore, some artists who like drawing like painting Is this argument valid or invalid ?

### Analytic

Suppose that f is analytic in the disc |z|<1, that f(0) = f'(0) = 0 and that |f(z)| &#8804; 1 for all z in the disc. Show that |f(z)| &#8804; |z^2|, for |z|<1. Hint: Schwarz's Lemma: Suppose that f is analytic in the disc |z|<1, that f(0) = 0 and that |f(z)| &#8804; 1 for all z in the disc. Then |f(z)| &#8804; |z|, for |z|

### 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

I am posting formula called Runge Kutta Fourth Order. This formula need to be written in a matlab program. I attached sample program how to generate euler formula in mathlab. Based on this program, can any one send matlab program for Runga Kutta Fourth Order? See attached files for full problem description.

### 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. ---

### Properties of Equalities

M1=35 and the m2=35. What property justifies the conclusion: m1=m2?

### Resolving at a Given Point

Require resolving at B to be worked with full solution and explanation please See attached.

### Statically determine a plane structure.

Statically determine a plane structure. Please see the attached file for the fully formatted problem and an example.

### Analytic Functions; Harmonic Functions; Laplace's Equation

5. Let the function ... be analytic in a domain D that does not include the origin ... 13. ... state why the functions ... are harmonic in D and why ... is in face, a harmonic conjugate 11. ... Why must this satisfy Laplace's equation? Please see attachment for complete questions. Thanks.

### 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

### Two lines intersect one another at the point (-4,1). Find their equations.

Two lines intersect one another at the point (-4,1), and create two triangles in the first and third quadrants (see figure). The area of both triangles is equal to 1. Find the equations of the intersecting lines.

### 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

### Tile a plane with n-gons.

Is it possible to tile a plane with (a) regular 5-gons and regular 6-gons? (B) regular 5-gons, regular 6-gons, and triangles? (c) regular 5-gons, regular 6-gons, and regular triangles?

### Find the vertex focus, and directrix of the parabola

1-Find the vertex focus, and directrex of the parabola= 1-x2= 2 (x+y) 2-Write an equation of the vertical parabola that contains (-1,2) with focus (3,1) and which opens upward. 3-Write an equation for the parabola that has vertex (1,2) and its axis is parallel to the x-axis, passes through the point (13,4) 4-Find all e

### Understanding solid figures.

What are solid figures?