# What is Analytic geometry: Definition and 49 Discussions

In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry.
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space (three dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom.

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1. ### I Converse of focus-directrix property of conic sections

In my recent study of Conic Sections, I have come across several proofs (many of those comprise Dandelin spheres) showing that the cross-section of a cone indeed follows the focus-directrix property: "For a section of a cone, the distance from a fixed point (the focus) is proportional to the...
2. ### I Expressing any given point on plane with one unique number

Currently, as far as I know, the two main ways to express any given point on a plane is through either cartesian plane or polar coordinates. Both of which requires an ordered pair of two numbers to express a point. However, I wonder if there exists such a system that could express any given...
3. ### Minimizing the friction of the end of a lever against the top of a moving piston

I'm trying to design a simple mechanism in CAD and got stuck on this problem. Consider a lever (blue shape) that pivots at one end (blue circle). The other end rests on a piston (pink rectangle). In my application the lever is pushing the piston (the lever pushes through the full travel of a...
4. ### B A Simpler Way to Find the Shaded Area?

Consider the following scenario: Given that points ##M## and ##N## are the midpoints of their respective line segments, what would be the fastest way to determine what percentage of the squares total area is shaded purple? I managed to determine that the purple shaded area is ##5\text{%}##...
5. ### Analytic geometry easy but tricky problem

I started off by indicating ##p=-\frac{1}{m}## since it's perpendicular. The sum ##m+p## is now ##\frac{m^2-1}{m}##. Honestly, I can't go beyond that. The interceptions with the y-axis are of course unuseful, I tried algebraically intersecting the two lines but I came up with nothing... and I...
6. ### Proofs in analytic geometry and vector spaces.

I was just thinking, if is said to me demonstrate any geometry statement, can i open the vector in its vector's coordinates? I will say more about: For example, if is said to me: Proof the square's diagonals are orthogonal, how plausible is a proof like?: d1 = Diagonal one = (a,b,c) d2 =...
7. ### B Elementary analytic geometry textbook recommendation

Every explanation about scaling a 2D vector, or equivalently having a line segment PQ on cartesian plane and then find a point R on the line PQ satisfying PR/PQ = r (fixed given r) starts with that one specific case in the picture. A formula for the coordinates of R is then given for that case...
8. ### MHB Analytic geometry proof with triangle.

Point D divides side AC, of triangle ABC, so that |AD|: |DC| = 1:2. Prove that vectors \vec{BD} = 2/3 \vec{BA} + 1/3 \vec{BC}.
9. ### B Correlation between shifting graph of a function and shifting the axes

1.To shift the graph of a function : Vertical Shifts : ## y=f(x) +h## where the graph shifts ##k## units up if ##k## is positive and downwards when ##k## is negative. Horizontal Shifts : ##y=f(x+h)## where the graph shifts to the left by ##h## units when positive and to the right when ##h## is...
10. ### Implications of zero in the denominator when solving equations

My book states the following : ##{\frac{(y_1)-(y_2)}{(x_1)-(x_2)}}*\frac{(3+1)}{(0-0)} = -1## ...(1) ##\implies y_1 - y_2 = 0## (A) Is this a valid deduction? Context : The Problem :If ##A(0,-1)## and ##B(0,3)## are two opposite vertices of a square, then find the other two vertices...
11. ### Distinguishing between angular bisectors

Homework Statement :[/B] The following expression stands for the two angular bisectors for two lines :\frac{a_{1}x+b_{1}y+c_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}}}=\pm \frac{a_{2}x+b_{2}y+c_{2}}{\sqrt{a_{2}^{2}+b_{2}^{2}}}\qquad Homework Equations The equations for the two lines are : ##a_1x + b_1y +...
12. ### Formulas for Rotation and Translation

Given the coordinates ##P = (3,4)## , find the coordinates of ##P"(x',y')## when the origin is shifted to (1, –2), and the axes are rotated by 90° in the clockwise direction. I attempted to solve this problem using the following formulas : ##x = X + h## and ##y = Y + k## for translation of the...
13. ### MHB Calculus and Analytic Geometry

My question is not a math question. I know about the calculus sequence (CAL 1, 2 and 3). I plan to go through all 3 in time. There is no rush for me. However, I know there is a course by the title of Calculus and Analytic Geometry. I want to know when this course is given. Is it given after...
14. ### Distance between point coordinates in a straight line

Homework Statement Let A = (1,2,5) and B = (0,1,0). Determine a point P of the line AB such that ||\vec{PB}|| = 3||\vec{PA}||. Homework EquationsThe Attempt at a Solution Initially, writing the line in parametric form\vec{AB} = B - A = (0-1,1-2,0-5) = (-1,-1,-5)\\ \\ \Rightarrow \vec{v} =...
15. ### MHB Analytic Geometry: Lines Problem

In a triangle ABC: A(-2,7) and C(7,-5). The length of the altitude of AC is 5, and the length of the altitude of BC is the square root of 45. I wish to find the vertex B, given that it is below the line AC. I need your help, I have no idea how to approach this. Thank you in advance. I have...
16. ### Analytic geometry: equations of planes (checking answers)

Homework Statement Let S be a sphere with the equation ##(x-2)^2+y^2+z^2=2 ## and let p a line which satisfies the condition ## p \in (\Pi \cap \Sigma) ## where ##\Pi## and ##\Sigma## are planes with equations: ##\Pi :x+z=2## ##\Sigma: 5x-2z=3## a) Show that S and p have exactly one common...
17. ### MHB Analytic Geometry: Confused on How to Solve Problem

I am actually very confused on how to solve the problem. Do I just find the distance between the lines? How do I incorporate the 500 Newtons into the problem? Really confused.
18. ### Proof of image formation property of spherical mirrors

Hi. I'm trying to proof the image formation property of a concave spherical mirror. I know you can do this easily with a particular choice of rays (namely one that hits the vertex and one that passes through the center of the sphere) but I would like to show that a generic ray yields the same...
19. ### Find a function with given condition

Homework Statement Find a curve that passes through point A(2,0) such that the triangle which is defined with a tangent at arbitrary point M, axis Oy and secant \overline{OM} is isosceles. \overline{OM} is the base side of a triangle. 2. The attempt at a solution Function passes through point...
20. ### Finding the Area Bounded by Two Functions

Homework Statement Find area bounded by parabola y^2=2px,p\in\mathbb R and normal to parabola that closes an angle \alpha=\frac{3\pi}{4} with the positive Ox axis. Homework Equations -Area -Integration -Analytic geometry The Attempt at a Solution For p>0 we can find the normal to parabola...
21. ### Newton's view on Line generation

The following passage has been extracted from the John Stewart's English translated version of the "Sir Issac Newton's two Treatises: Of the Quadrature of Curves, and Analysis by equations of an infinite number of terms" http://archive.org/details/sirisaacNewtons00stewgoog: Here Newton...
22. ### Analytic geometry, proper order for learning and teaching it

I'm planning to write about analytical geometry, but I'm unsure about where to start. The course that I've taken begins with the definition of a vector, it then proceeds to develops all the vector algebra and all properties. After it reaches the cross product and triple product it begins to...
23. ### Help with Proving Ruled Surface Equation

Hello, I am studying for an analytic geometry final but I am totally lost for this problem... We didn't even cover this topic in class (my prof didn't show up for class for two weeks) and I have no clue on how to do it. If anyone could help I would appreciate it. Question: Prove that the...
24. ### Am I ready for Calculus and Analytic Geometry?

Hello all, I was accepted into a program that allows me to take a free college course and I was hoping to take a math/science course. I am most interested in Physics, but I don't know Calc so that isn't really an option, and I was thinking of taking Calculus and Analytic Geometry. However, I am...
25. ### Calc w/ analytic geometry test

So I am trying to test into the Calculus with Analytic Geometry class at my school. I've already taken the first AP Calculus class my senior year of high school (but wasn't particularly concerned with my grade, and never took the test, since it was an extra curricular class for me). Then I took...
26. ### Finding Slope and Y-Intercept for Points on a Line: A Simple Explanation

Hey so I have a math test tomorrow and I'm really good in math, I just need some small clarification, so here's my problem. V(-2,k) P(4,9) slope of 2/3 so I decided to find y=mx+b with P(4,9) because after I find B I could do the same with V(-2,k) so I did the equation and B came up 7...
27. ### MHB Looking for a good analytic geometry book

I am always entranced when I read in calculus books about various curves like lemniscates, cardioids, the spiral of Archimedes, the witch of Agnesi, and similar things. But in calculus books they are generally just little asides or relegated to the exercises. Presumably there is more to analytic...
28. ### Calculus w/ Analytic Geometry vs. Calculus for Engineers

Hello PF! After months of eye shopping, I couldn't help but join this awesome community myself. I am a high school senior entering college soon, who is very excited to take college-level physics! I've done some research of what classes I should take, and came upon a conflict of scheduling...
29. ### MHB Unknown_12's questions at Yahoo Answers regarding analytic geometry

Here are the questions: I have posted a link there to this thread so the OP can see my work.
30. ### MHB What is the equation of the extreme normal line for a parabola?

The curve in the figures above is the parabola $y=x^2$. Let us define a normal line as a line whose first quadrant intersection with the parabola is perpendicular to the parabola. Five normal lines are shown in the figures above. For a while, the $x$-coordinate of the second quadrant...
31. ### MHB Suggestions for Analytic Geometry books

Suggestions for Analytic Geometry books; What do you have in mind?:)
32. ### Analytic Geometry / Vectors - Find point with min distance to plane

Question: Find the point in the plane x+2y-3z=1 with minimum distance to point (1,1,1). Attempt at resolution: Well, techinically, I already have the means to solve this. I can find the distance between the point and the plane to be 1/sqrt(14) and then I can either solve for the sphere...
33. ### Analytic Geometry / Vectors - Finding an Angle Bissector

Question: What are the vector, parametric and symmetric equations of the angle bissector of angle ∠ABC, given that A=(1,2,3), B=(3,4,5) and C=(6,7,0). Attempt at resolution: Well, I defined some D=(d_1,d_2,d_3) to be a point in the angle bissector, and two lines r:X=(3+2a,4+2a,5+2a) and...
34. ### A Historical Look at Analytic Geometry

Hello, Analytic geometry has provided us with such profound tools for thinking that it is hard to imagine what thinking must have been like before we had such tools. Two particular developers of these tools are Pierre de Fermat and Renee Descartes in 17th century France. I would like to...
35. ### MHB How to Solve Vector Equations and Find Distances in Analytic Geometry?

I'm trying to help a friend but I don't remember any of this, so it'd help us both. Book recommendations are also welcomed. Given the vector equations r: \begin{cases} x = 2 - \lambda, \\ y = 1 + 3 \lambda, \\ z = 1 + \lambda, \end{cases} \text{ and } s: \begin{cases} x = 1+t, \\ y = 3+4t, \\...
36. ### Analytic Geometry Question (equation of a circle)

Homework Statement Find the equation of the circle whose centre lies on the x-axis and which passes through points A (6,0) and B (0,10). Homework Equations The Attempt at a Solution I drew a diagram of the circle and determined that the line AB has gradient 5/3. Its perpendicular bisector...
37. ### How Do You Derive the Equation of an Ellipse for a Semi-Elliptical Ceiling?

Homework Statement . 1. a hall that is 10 ft. wide has a ceiling that is a semi-ellipse. the ceiling is 10 ft high at the sides and 12 ft high in the center find its equation with the x-axis horizontal and the origin at the center of the ellipse. Homework Equations x^2/a^2 + y^2/b^2 = 1...
38. ### Where Can I Find Good Analytic Geometry Textbooks Online?

Hello! I'm looking for a good analytic geometry textbook. I'm studying circumferences and sphere and I'm using the book Analytic Geometry: A vector approach from Wexler but I find this book quite bad. I want some suggestions! Thank you.
39. ### Analytic Geometry World Problem

Homework Statement Points S and T lie on the line 2x+y=3. If the length ST is 13 units, and the coordinates of S are (2, -1), determine all possible coordinate pairs for T, correct to two decimal places. For simplicity, determine the coordinates for T, relative to S from the left.Homework...
40. ### Understanding Tangents in Analytic Geometry

Okay so I am reading a book on analytic geometry because my algebra class is starting to really bore me, and I ran across this: slope = m = -1/2 (easy enouph) but then it states "since the slope is the tangent of the angle of inclination θ, we have: tan θ = -1/2 okay so I guess my...
41. ### Calculus with Analytic Geometry 1 Book?

I am a sophomore at Penn State and I am taking Calculus for the second time, I didn't fail the first time, but I didn't want to take the chance I might get a D so I dropped it. Anyhow I am looking a great Math book to learn from while I am learning it again in class. We are using Calculus: Early...
42. ### Analytic geometry question: Finding the Parameters of an Ellipse

Find the vertices, foci, and eccentricity of the ellipse. 1/2x^2 + 1/8y^2 = 1/4 i'm a little confused on how to clear the fraction and set the equation up to make it equal 1. I tried to divide everything by 4 to clear it but 8 can't divide into 4 and if I use a higher number than 4 the...
43. ### Place of Analytic geometry in modern undergraduate curriculum

Hello. I am a freshmen student in mathematics at Moscow State University (in Russia) and I'm confused with placing the subject called "analytic geometry" into the system of mathematical knowledge (if you will). We had an analytic geometry course in fall; now we are having a course in linear...
44. ### Find Coordinates of Point P in Analytic Geometry

Homework Statement In the figure, the orange line and the green line are perpendicular and intersect on the x-axis.Find the coordinates of P. http://tinypic.com/r/2wn7m2v/3 go here for the image please. Homework Equations slope-point formula, m1*m2=-1 FOR PERPENDICULAR LINES The...
45. ### Analytic Geometry - finding the normal vector at a point

I'm struggling to find the method to use when trying to get the normal vector n to a (flat) surface at a specific point. Every textbook I've researched has the same scenerio, where they show how to get a plane given a point and normal vector, but I have the opposite problem, where I know the...
46. ### How to calculate angle in analytic geometry

Homework Statement http://img196.imageshack.us/img196/5391/29069034.jpg Homework Equations only analytical methods The Attempt at a Solution I found k to be =2 I don't have any idea how to calculate angle A. Please need some help. Thank You.
47. ### Analytic Geometry Problem - find a line in space meeting some conditions

Hi all! I've appreciated (and lurked :-) this forum for a while. While waiting to find a question I'm able to answer, it's time for my first post (hopefully the first of many). Here's the problem: I'm trying to teach myself euclidean geometry and a found a tough (for me) question to answer in...
48. ### Finding the Center and Radius of a Circle: Analytic Geometry I

Homework Statement Find the centre and radius of the circle x²+y²-4x+2y+6=0 I have the solution. The circle is no defined because r² = -1 is impossible. But... how do I even DO that equation to get the answer -1?!
49. ### Analytic Geometry: Proving R Lies on Ellipse

Hi all. I have a analytic geometry question that I need a bit of help with. consider the concentric circles with the equations: x^2 + y^2 = 9 and x^2 +y^2 = 4 A radius from the center O intersects the inner circle at P and the outer circle at Q. The line parallel to the x-axis through P...