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.

Filter by
Sorted by
Tagged with
3
votes
0answers
62 views

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 ...
1
vote
0answers
23 views

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 ...
0
votes
4answers
34 views

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 ...
-4
votes
1answer
54 views

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 $a, (3/2)a, b$....
1
vote
0answers
26 views

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 ...
0
votes
0answers
10 views

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,...
-3
votes
0answers
43 views

$G$ 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 $F$ ...
1
vote
1answer
28 views

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 ...
0
votes
1answer
41 views

A curios fact on tangent line on circumferences.

I have the following point $A(-1;1)$ and the line $s:x-2y+3=0$. Consider the general equation of the circumference: $$x^2+y^2+ax+by+c=0.$$ I applied the following transformation rule: $$ x^2 \to x \...
-1
votes
2answers
49 views

Partition of a triangle into equal areas

A square piece of toast ABCD of side length 1 and centre o is cut in half to form two equal pieces ABC and CDA. If the triangle ABC has to be cut into two parts of equal area, one would usually cut ...
0
votes
1answer
24 views

Question about a series of distance preserving transformations on points

I have a problem that asks me to Find all length preserving transformations of the plane that send point A to point A’ and point B to point B’ where: $A=(0,1), B=(1,1), A’=(3,2), B’=(3- \frac{\sqrt3}...
0
votes
1answer
32 views

If two lines are perpendicular to each other and a third line bisects the right angle, then what would be the equation of that bisector?

If two lines are perpendicular to each other and a third line bisects the right angle, then what would be the equation of that bisector? I mean, I know the equations for acute and obtuse angle as ...
1
vote
0answers
47 views

Co-ordinates of the point on a sphere which is closest and furthest to another point

I just need some help with this question. Let $S$ be the sphere of radius $2$ centred at the origin. Find the co-ordinates of the point on $S$ which is closest to and furthest to the point with co-...
-1
votes
2answers
34 views

Finding the equation of the line through the points of tangency from point $(8,10)$ to circle $(x+12)^2 + (y+5)^2 = 225$

$A$ and $B$ are the points of tangency of tangents drawn from $P(8,10)$ to the circle $(x+12)^2 + (y+5)^2 = 225$. Find the equation of the line $AB$. Since $AB\perp{CP}$, where $C$ is center of ...
0
votes
3answers
53 views

Estimating the sine of the angles between vectors

Let $\{s_n\}$ be a sequence of unit vectors in $\mathbb{R}^2$ converging to some $s\in\mathbb{R}^2$. I want to show that $$\sin\measuredangle(s_n,s)\le \sum_{m=n}^\infty\sin\measuredangle(s_m,s_{m+1})$...
2
votes
1answer
55 views

A property of parallelism in a figure formed by an ellipse and a circle with same area

Consider a circle of radius $r$ and an ellipse of semiaxes $a>b$, both centered at the origin $O$, with area of the circle and the ellipse being equal, such as in the Figure. The ellipse has an ...
2
votes
2answers
82 views

Prove that three circumferences in a triangle intersect in a point

We've been given this exercise by our professor as an optional one, but I don't know the correct way to tackle it. Let $T=\triangle\{A,B,C\}$ be a triangle. Consider three points $A', B', C'$ so that ...
0
votes
0answers
24 views

The rotation matrix of a triangle in the 3D space

Consider a triangle in the 3D polar coordinate system denoted by $T=\{O,(r_1,θ_1,φ_1),(r_2,θ_2,φ_2)\}$, where $O$ is the origin. Rotate this around $O$ and we obtain a new triangle $T'=\{O,(r_1,θ_1',...
0
votes
1answer
42 views

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 ...
0
votes
2answers
24 views

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.
1
vote
2answers
32 views

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 ...
-1
votes
0answers
28 views

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&...
0
votes
0answers
31 views

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 ...
0
votes
1answer
45 views

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 ...
2
votes
1answer
35 views

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 ...
-3
votes
1answer
33 views

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 ...
0
votes
0answers
17 views

Rewriting path difference $d_2 - d_1 $ with simple trig.

Consider a system where waves propagated through an aperture with a size of $a$ (illustrate in the figure below). The path difference $d_2 - d_1$ can be rewrite to: $(\sqrt{d^2+a^2}-d) \approx \frac{a^...
1
vote
1answer
17 views

Extension of linear dependency of polynomials defined on analytical manifolds to the ambient space

Let $M \subset \mathbb{R}^n$ be an analytic manifold. Let $p_i$, $i=1,\cdots,m$ be $m$ polynomials on reals defined in $\mathbb{R}^n$ and $m < n$. We assume that there is a linear dependency of $...
0
votes
0answers
11 views

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 ...
-2
votes
1answer
65 views

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 ...
0
votes
0answers
20 views

Busemann functions and inequalities

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}{...
1
vote
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 ...
0
votes
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 $S$. ...
1
vote
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 $D_r$ ...
5
votes
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 $B'C'$. $\frac{\angle BAD}{\angle CAD}=\frac{\angle B'A'D'}{\angle C'A'D'}$, $AB=A'B'$...
0
votes
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 ...
0
votes
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$. Find the parametric equation of the curve$...
0
votes
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 ...
1
vote
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) ...
0
votes
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 ...
0
votes
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 ...
0
votes
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=-\...
4
votes
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 ...
2
votes
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)=...
5
votes
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
0answers
19 views

Homothety of general figures

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), ...

1
2 3 4 5
111