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Questions tagged [analytic-geometry]

Questions on the use of algebraic techniques for proving geometric facts. Analytic Geometry is a branch of algebra that is used to model geometric objects - points, (straight) lines, and circles being the most basic of these. It is concerned with defining and representing geometrical shapes in a numerical way.

Analytic geometry, also called coordinate geometry, mathematical subject in which algebraic symbolism and methods are used to represent and solve problems in geometry.

The importance of analytic geometry is that it establishes a correspondence between geometric curves and algebraic equations. This correspondence makes it possible to reformulate problems in geometry as equivalent problems in algebra, and vice versa; the methods of either subject can then be used to solve problems in the other.

Analytic geometry was introduced by René Descartes in $1637$ and was of fundamental importance in the development of the calculus by Sir Isaac Newton and G. W. Leibniz in the late $17^{th}$ cent. More recently it has served as the basis for the modern development and exploitation of algebraic geometry.

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I need help solving the equation for my revision class

I need to solve the radius and the center point. $$ x^2+y^2=6x-2y-8 $$ Under this is my attempt of solving it $$ x^2-6x+y^2+2y=-8 $$ $$ x^2-6x+9+y^2\ +2x+1=-8+1+9 $$ $$ \left(x+3\right)^2+\left(y-1\right)^2=2^2 $$ On 2nd I added the integers to both…
Elias
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Hyperbola given by the midpoint of two LI vectors

Here is the problem: Let u and v be two LI vectors in $\mathbb{R}^3$. a) Prove that the area of the triangle with sides $\lambda \overrightarrow{u}$ and $\frac{1}{\lambda}\overrightarrow{v}$ doesn't depend on $\lambda$. b) Prove that the midpoints…
Igor Souza
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Trying to understand this solution for a high-school coordinate geometry problem better

$\newcommand{\lrp}[1]{\left(#1\right)}$ The following is a high school level problem in coordinate geometry and the purpose of this post is to understand the solution better or discover other solutions or both. Problem. Show that all chords of the…
caffeinemachine
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Find the asymptotes of $xy^2 - y^2 - x^3 = 0$

I am lost with this question because there is no 3rd degree $y$ term and no 2nd degree $x$ term. Can $y - x - 1/2 = 0$ and $y + x + 1/2 = 0$ be its asymptotes? or $y - x - 1/3 = 0$ and $y + x + 1/2 = 0$
falikammarahmed23
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Check whether the three vectors $A(2,-1,2),B(1,2,-3),C(3,-4,7) $ are in the same plane

I want to check if three vectors are in the same plane, the vectors being $$A(2,-1,2),B(1,2,-3),C(3,-4,7). $$ What I did so far is to create vector $AB ( -1,3,-5)$ and build the plane equation with the point $A$ $$-1(x-2)+3(y+1)-5(z-2)=0$$ and…
Ofir Attia
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Klein Model Geometry

I have two questions: What is the special case of parallels when the Klein model will not create a perpendicular? II) From point P, drop a perpendicular Q with line l. Consider the hyperbolic angle
qjl
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Find the vector by the following criteria

I want to find the vector that meets the following: $$X\parallel (2,1,-1)$$ $$X*(2,1,-1)=3$$ what I did so far is : $$2x+y+z=3$$ I know that parallel vectors the angle is $180$ or $0$. how to continue from here? Note: X is vector. Thanks!
Ofir Attia
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Finding the sum of $5$ angles given equivalent division of a side

Let $ABC$ be an isosceles triangle with $AB = AC.$ Point $D$ lies on segment $AB$ so that $AD = AB/6.$ Points $E_1, E_2, E_3, E_4,$ and $E_5$ lie on segment $BC$, in this order from $B$ to $C$, and they divide it into six equal parts. Find $$\angle…
questionasker
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Proving four points give a parallelogram (is two vectors being parallel enough?)

Let's say that I am given four points and I want to prove they form a parallelogram. What I usually see in books is that they calculate $\vec{AB}$ and show that it is equal to $\vec{DC}$, but they also show that $\vec{BC}$ is equal to…
bru1987
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An ellipse has foci at $(1, -1)$ and $(2, -1)$ and tangent $x+y-5=0$. Find the point where the tangent touches the ellipse.

An ellipse has foci at $(1, -1)$ and $(2, -1)$ and tangent $x+y-5=0$. Find the point where the tangent touches the ellipse. Here is a procedure how to do it analiticaly. If $T(x_0,y_0)$ is a touching point, then $x_0+y_0=5$ The equation of…
nonuser
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Proving the angle trisection identity of a limaçon

My attempt to prove the angle trisection identity of limaçons. In the diagram below, I drew a limaçon trisectrix having polar equation $r=a+2a\cos\theta$ (In this case, $a=1$). Suppose $O(0,0), A(a,0), B(3a,0)$. Then, draw a line which has cartesian…
E. Huang
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Why are sheafs of planes linear combinations of two other planes

In most books, it is simply stated that the sheaf of planes can be described as a linear combination of two planes, but I cannot alight upon a simple explanation as to why. I can see that a symmetric line equation in 3-D can be expressed as three…
JerryFrog
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Necessary equation for envelopes

Consider a family of curves parameterised by $t$, where each member of the family is described by $$F(t,x,y) = 0.$$ Define an envelope of the family to be a curve $E$ such that every point on $E$ is tangent to some member of the family. When do we…
Mr. Chip
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how to dot product two vectors with different planes?

how to dot product two vectors with different planes? I have vectors $A$,$B$ and $C$, vectors $A$ and $B$ is on $xy$ plane while vector $C$ is on $xz$ plane. I need to find the dot product of $A.C$ how should I do that? my book says that dot product…
nicy12
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What is the importance of fixing an origin in a coordinate system?

In analytic geometry, a coordinate system is defined as as pair $(O, \mathcal{B})$ consisting of a point $O$ in the space, called the origin, and a basis $\mathcal{B}=(\overrightarrow{b}_1, \overrightarrow{b}_2, \overrightarrow{b}_3)$ for the space…
PtF
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