# 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.

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### Geometric quantity related to $a^3 + b^3 + c^3$

The following geometric proof of the Pythagoras theorem relies on the fact that one can cut out 4 right angled triangles (of area $\frac12ab$ each) out of a square of side length $a+b$ to obtain ...
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### Projection of moving point onto static curve and respective velocities / Frenet Coordinates

Consider a curve in 2D $\vec{p}(s)$ parameterized by arclength $s$ and the usual local coordinate system on the curve (Frenet Frame with unit vectors $\vec{n}(s)\perp\vec{t}(s)$, no torsion, curvature ...
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### Find an equation for the line tangent to the graph of $f^{-1}$ at the point $(3,1)$ if $f(x)=x^3+2x^2-x+1$

Find an equation for the line tangent to the graph of $f^{-1}$ at the point $(3,1)$ if $f(x)=x^3+2x^2-x+1$ ok, so I know that I need to take the derivative of f(x). $f'(x)=3x^2+4x-1$ The inverse ...
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### Just did my first A level exam, wondering how to solve these two problems: [closed]

I screwed up 4 questions and I'm probably gonna lose about 20 marks (so I'll get something like 60/75) The first question is an arithmetic one. The arithmetic sequence has first 3 terms $a, (3/2)a, b$....
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### Lorentz Transformation Geometric Interpretation

So, I was recently trying to understand special relativity and from my understanding the Lorentz transformation can be framed mathematically as follows (we are assuming a reference frame moving at ...
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### Calculating the azimuthal angle in different coordinate systems

For a right-handed coordinate system, atan2(y,x) calculates the azimuthal angle. What is it for a left-handed coordinate system, i.e., with the z axis pointing down,...
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### $G$ is the centroid of $\triangle{ABC}$. Perpendiculars from vertices $A,B,C$ meet on the sides $BC,CA$ and $AB$ at $D,E$ and $F$ respectively

Hints: This is from the chapter solution of Traingle Question: $G$ is the centroid of $\triangle{ABC}$. Perpendicular lines from vertices $A,B,C$ meet on the sides $BC,CA$ and $AB$ at $D,E$ and $F$ ...
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### Define the open set {$O_\alpha$} that covers the $n$-dimensional sphere and the charts $\psi_\alpha:O_\alpha \rightarrow U_\alpha \subset R^n$.

That is precisely the question. I thought I would just take n-dimensional spherical coordinates, but somehow that doesn't seem to work. The tasks that build on this make it impossible. Maybe I'm just ...
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### Find the probability: Two points are selected randomly on a line of length L

Please help me with this one. Two points (B and C) are selected randomly on a line of length L. Find the probability that the segment BC has a length less than L / 4. It is assumed that the ...
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### Parabola above $x$-axis

Why is the quadratic(or maybe other degrees) polynomial $ax^2+bx+c$ with $a$ positive has a parabola having both, its ends always above the $x$-axis? I am not getting the logic behind it.
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### How will we find the equation of AC?

The vertex A of $\triangle$ABC is $(3,-1)$. The equations of median BE and angle bisector CF are $x-4y+10=0$ and $6x+10y-59=0$, respectively. What will be the equation of AC? I tried using the ...
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### The range of values of θ,θϵ[0,2π] for which (cosθ,sinθ) lies inside the triangle formed by x + y = 2, x − y = 1and 6x + 2y − √10 ​=0 .

not getting any hint of how can I get the range of those points is there any short method of doing so I have done it like plotted the graph and then I found out that the line "6x + 2y − √10 ​=0&...
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### Problem on square with vertex on a hyperbola

Given the equation $\frac{x^2} {3}-\frac {y^2} {12}=1$ and a square with vertices on the hyperbola and sides parallel to the axis prove that every vertex has this property: Let $A$ be the vertex, and ...
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### Horizontal distance between circle and ellipse centered at origin

The figure below shows a circle of radius $r$, centered at the origin of cartesian coordinates, and an inclined ellipse also centered at the origin, with semiaxes $a$ and $b$. The area of the circle ...
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### Given point $A$ in the interior of a circle and point $B$ outside the same circle, prove that there is a point in the circle in $\overline{AB}$

I want to run away from continuity/analysis and atempt to prove it using euclidean geometry theorems. I do know that the interior of a circle is a convex region. But I lack a smart way to prove the ...
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### The 2D transformations [closed]

I would like to ask a question relating to the transformations in the 2D plane. What are the transformations 2D which preserve the X coordinates and change the Y coordinates which causes lines that ...
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### With Geogebra, how to defne a position vector in a moving frame $(O', \vec{i'},\vec{j'})$ in order to have this vector placed at $O'$,not at$O=(0,0)$.

Context of my question : understanding some basic cinematic equations by creating a toy model of moving 2D relative referential in Geogebra . In Geogebra, it's possible to create a relative 2D ...
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### Why does all points $(x,y)$ satisfying $ax+by=c$ stay on a straight line? [duplicate]

We know that points $(x_i,y_i)$ which satisfy the equation $ax_i+by_i=c$ lie on the same straight line. I understand that all points on this line satisfy the equation, but how do we ensure that all ...
Let $D$ is the hyperbolic unit disk. Let $\alpha,\,\beta\in S^1$, where $S^1$ is the boundary of $D$. Let $w\in D$. I know that Busemann function for hyperbolic disk is $$B(w,\alpha)=\ln\frac{1-|w|^2}{... 4answers 71 views ### what is a^2+9=b^2+16=1+(a+b)^2 solve for a,b [closed] This is for a geometry question, and through a construction arrived at this equation. I could not solve it and after plugging it into wolfram got the correct answer but can anyone show a method for ... 1answer 52 views ### Find the Locus of the foot of the perpendicular. Consider the tangent planes to the surface S: \frac{x^2}{2}+y^2+z^2=1 that passing through the point P(1,1,1), then draw the perpendicular to the tangent plane from the centre of the surface S. ... 0answers 87 views ### Equation of a line arises as limit of the equation of two circles in the complex numbers Let u, v \in \mathbb{C}. Consider the circles centered at u, v such that they intersect exactly at the point \frac{u+v}{2}, i.e. C_u = \partial D_r(u), C_v = \partial D_r(v) where D_r ... 1answer 102 views ### Prove ABC,A'B'C' are congruent:D is on BC,D' is on B'C', \angle BAD \angle CAD= \angle B'A'D' \angle C'A'D', AB=A'B', AC=A'C', AD=A'D' In \triangle ABC and \triangle A'B'C', D is a point on line segment BC and D' is a point on line segment B'C'. \frac{\angle BAD}{\angle CAD}=\frac{\angle B'A'D'}{\angle C'A'D'}, AB=A'B'... 0answers 234 views ### Is there a general theorem for lines intersecting in R^2_{++} space? Im wondering if there is a general theorem which discusses whether or not two lines that intersect in the \mathbb{R}_{++}^2. This is visualized in the graph below. If there is a broader theorem ... 1answer 24 views ### Find the parametric equation of the curve Let\ R\ be the radius of curvature of the plane curve\ γ,\ α\ be the angle between the constant vector and the current tangent vector of the curve\ γ. Find the parametric equation of the curve... 1answer 29 views ### Determine the equation of a a plane tangent at a hyperboloid of one sheet in a point M. Prove that this tangent plane cuts the surface after two lines Determine the equation of a plane tangent at a hyperboloid of one sheet \frac{x^2}{4}+\frac{y^2}{9}-\frac{z^2}{1}=1 in a point M (2,3,1) . Prove that this tangent plane cuts the surface after two ... 1answer 31 views ### How to show that congruent chords are equidistant from circle? I tried like that but did not get the required result. Let the circle equation with center C(-f,-g) is: x^2+y^2+2gx+2fy+c=0.................(1) ... 1answer 32 views ### How to find this line length and targeted point coordinates based on other points? First of all I'm beginner in "advanced" math. For this reason I don't know how to compute this problem. Consider we have a generic rectangle with width W and height H. Also, consider that ... 1answer 44 views ### Question related to ellipse Let 'd' be the perpendicular distance from the centre of the ellipse \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1 to the tangent drawn at at point 'P' on the ellipse. If F_1 and F_2 are ... 0answers 26 views ### Writing equation in polar coordinates for tangent circle How can I write the equation for this tangent circle? Fundamental circle is r=3\sin(\theta) and I also find the tangent line for \theta = \frac{\pi}{3} And the tangent line to the circle is: y=-\... 3answers 90 views ### Find the slope of the line such that the area of triangle formed inside the circle is maximum Here is the question. It states: A circle of radius 1 unit touches positive x -axis and positive y -axis at A and B respectively. A variable line passing through origin intersects the circle in ... 1answer 84 views ### Doubts about elementary geometric proof in Arnolds' “Lectures for young mathematicians”? I'm reading Arnolds' "Lectures for young mathematicians", there is this proof: I am a bit confused about two things: In the first line of the centralized equations, they write: area(OACB)=... 0answers 101 views ### Proving the existence of the Euler line using methods from Coordinate Geometry. I saw a video by Salman Khan, in which he gave a proof of existence of the Euler Line. He proved that the circumcenter, orthocenter and centroid of a triangle are collinear, and used normal geometry ... 1answer 33 views ### Find all integer points at distance d from line segment (0, b) I'm reading a scientific paper on integer linear programming and trying to understand a specific part of it. There is a point  b \in \mathbb Z^m  and a set \mathcal S that consist of all points x ... 4answers 77 views ### 2x^2 + xy + 2y^2 = 0 a pair of straight lines Is the following equation a Pair of Straight lines ? 2x^2 + xy + 2y^2 = 0 I can see h^2 - ab is negative. I do not think it will be a Pair of Straight lines. Then what is it ? Can anyone please ... 2answers 41 views ### [Computational Geometry]How to find the area between intersecting circles I have a bunch of circles that intersect. For instance, in the diagram below, there are four points A, B, C and D located at (0,0), (0,1), (1,1) and (1,0). There are a total of 6 edges, one ... 0answers 61 views ### What manifold is described by ad=bc? Consider the set in \mathbb{R}_+^4,$$ S = \{ (a,b,c,d)\; | \; ad=bc,\; a,b,c,d\geq0\}. What is this manifold? (I.e. does it have a name?) For starters, I believe $S$ is smooth, connected and ...
Is there an equivalent statement (other than finding the center of homothecy) which assures homothecy? I have the following problem. I have two figures in a plane, $A$ and $B$ (both convex), ...