Tuesday, December 5, 2023

# Analytic Geometry Problems And Solutions

## Analytic Geometry 2 Problems

Analytic Geometry – Distance Formula

1st problem : find the equation of the straight line having slope \$m\$ passing through the point \$\$. What are the coordinates of the point of intersection of this line with the y-axis?2nd problem : find an equation of the line passing through the point \$\$ with a negative slope and forms with the coordinates axis a triangle with area \$10\$ sq. units. Please I want answers with steps.

Thank you!

• \$\begingroup\$It is much easier to google: equation of a line.\$\endgroup\$Nov 9, 2012 at 5:33
• 3\$\begingroup\$So, what do you know about equations of lines? Hard to answer the questions in a helpful way without knowing that.\$\endgroup\$ Gerry MyersonNov 9, 2012 at 5:35
• \$\begingroup\$i have only 15 years old and i have try to solve them more than 5 times !\$\endgroup\$Nov 9, 2012 at 5:36

## Introduction And Historical Remarks

In mathematical deformation theory one studies how an object in a certain category of spaces can be varied as a function of the points of a parameter space. In other words, deformation theory thus deals with the structure of families of objects like varieties, singularities, vector bundles, coherent sheaves, algebras, or differentiable maps. Deformation problems appear in various areas of mathematics, in particular in algebra, algebraic and analytic geometry, and mathematical physics. According to Deligne, there is a common philosophy behind all deformation problems in characteristic zero. It is the goal of this survey to explain this point of view. Moreover, we will provide several examples with relevance for mathematical physics.

Historically, modern deformation theory has its roots in the work of Grothendieck, Artin, Quillen, Schlessinger, KodairaSpencer, Kuranishi, Deligne, Grauert, Gerstenhaber, and Arnold. The application of deformation methods to quantization theory goes back to BayenFlatoFronsdalLichnerowiczSternheimer, and has led to the concept of a star product on symplectic and Poisson manifolds. The existence of such star products has been proved by de WildeLecomte and Fedosov for symplectic and by Kontsevich for Poisson manifolds.

Sheldon M. Ross, in, 2017

## What Is Analytic Geometry

Analytic geometry is that branch of Algebra in which the position of the point on the plane can be located using an ordered pair of numbers called as Coordinates. This is also called coordinate geometry or the Cartesian geometry. Analytic geometry is a contradiction to the synthetic geometry, where there is no use of coordinates or formulas. It is considered axiom or assumptions, to solve the problems. But in analytic geometry, it defines the geometrical objects using the local coordinates. It also uses algebra to define this geometry.

Coordinate geometry has its use in both two dimensional and three-dimensional geometry. It is used to represent geometrical shapes. Let us learn the terminologies used in analytic geometry, such as

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## Analytic Geometry In Three Dimensions

In this, we consider triples which are real numbers and call this set as three- dimensional number space and denote it by R. All the elements in the triple are called coordinates.

Lets see how three-dimensional number space is represented on a geometric space.

In three-dimensional space, we consider three mutually perpendicular lines intersecting in a point O. these lines are designated coordinate axes, starting from 0, and identical number scales are set up on each of them.

Learn more on Coordinate Geometry in Two Dimensional Plane

## Putting Geometry To Work

Analytic geometry or coordinate geometry is geometry with numbers. In analytic geometry, vertices and special points have coordinates & #8212 in the 2D plane, in 3D space, and so on. Curves are represented by equations. For example, the graph of x2 + y2 = 1 is a circle in the x-y plane.

We manipulate these coordinates and equations to change geometric figures, explore their properties and create new forms. Many of the foundational ideas in this section have been covered in other sections on distance, slope, midpoint and conic sections.

#### Ideas Reality

Analytic geometry is how we translate engineering, architectural, artistic and other ideas into a language that builders, machinists and machines can use to physically create the things of which we dream.

The Boeing 777 aircraft, ubiquitous in the skies now as a passenger carrier, was the first such plane that was entirely built and tested on computers before a prototype was manufactured.

Imagine trying to manufacture the complexly-shaped turbofan blades of the GE turbine engine . Each contains complicated curves in each of the three dimensions that must be precisely formed in hard metals that are likely difficult to work with.

Parts like these are made by programmed machines, and that programming is mathematical coordinates and equations that specify every feature of the part, translated from an engineer’s mind to his/her computer to the machine.

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## Secularization And The Divine

During the Renaissance there was still in Western civilization a shared core of faith. The Scientific Revolution sowed the seeds of secularization. Conflicts between believers were replaced by debates between believers and non-believers. The different views on mathematics and the divine that were expressed in this period reflect this radical development. In Chapter 21, Donald Adamson discusses the ideas of Blaise Pascal. To many the new mechanistic philosophy represented a threat to religion. Pascal tried to safeguard religion by strictly separating the scientific domain, with its geometric approach from the religious domain. In Chapter 23 Philip Beeley and Siegmund Probst treat the views of John Wallis . In the confrontation between theology and science Wallis protected religion by keeping his work in mathematics apart from his theological work. In this way he could handle infinity in mathematics in an unconstrained way and at the same time, on a theological level, attack Hobbes who had claimed that there was no argument to prove that the world has a beginning.

Spinoza has been interpreted in different ways. Some have interpreted his pantheism and his emphasis on the amor dei intellectualis as a form of mysticism. Others view him as an atheist, because he abolished God as a creator. Jonathan Israel, for example, considers Spinoza as the crucial representative of what he calls the Radical Enlightenment. Israel writes:

M.J. Pflaum, in, 2006