Questions on the use of algebraic techniques to prove geometric theorems.

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3answers
21 views

Proving that the lines cut the coordinate axes in concylic points

The lines $2x +3y +19 = 0$ and $9x+6y-17 = 0$ cut the coordinate axes in concyclic points.What would be the fastest method to prove it manually?Is it possible to prove the statement without having to ...
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1answer
19 views

To find the centre of the inner circle that is tangent to the unit circle and the x-axis

We have a unit circle $C:x^2+y^2=1$. Let $l:y=m(x+1)$. We consider a circle $C'$ at a centre on $l$ that is inscribed to an upper semi-circle, i.e., a circle that is tangent to the circle $C$ and the ...
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0answers
11 views

Pinhole projection of the center of a 3D circle

Consider the pinhole projection of a 3D circle. The projection I am considering is a pinhole camera projection which has a fully known calibration. The projection of a 3D circle will be an ellipse, ...
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0answers
8 views

Split a map into roughly equal sections directionally and put points in it

I have a 16000 x 9000 grid map and I want to split it into x sections that are preferably of equal size. Then I want to place points on each section are centers of circles with a 2200 unit radius and ...
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0answers
64 views

Number of $N$ formed from the set of points

Given $k$ points on 2d plane, I need to find the number of $N$ shaped figures from these $k$ points. lets consider four different points from the set and name them $A$, $B$, $C$, and $D$ (in that ...
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5answers
1k views

How to tell whether a point is to the right or left side of a line

I have a line equation in the form ax+by+c=0 and a point p(x,y).How can I determine on which side of the line the point is located?
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1answer
37 views

Ellipse and horizontal lines

Let's imagine that we have an ellipse described by the known equation $v^TAv=0$, (Link_1) where $v=[x \ y \ 1]^T$ (it can be a skew one in a general case). Then we have all possible horizontal lines ...
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2answers
125 views

3-D Geometry Problem. Find a curve which touches the straight line.

If two perpendicular tangent planes to paraboloid $x^{2}+y^{2}=2z$ internsects in a straight line in the plane $x=0$, obtain the curve to which the straight line touches. I don't know how to ...
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3answers
14k views

How to find shortest distance between two skew lines in 3D?

If given 2 lines $\alpha$ and $\beta$, that are created by 2 points: A and B 2 plane intersection I want to find shortest distance between them. $$\left\{\begin{array}{c} P_1=x_1X+y_1Y+z_1Z+C=0 ...
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1answer
9 views

Find point where a line of multiple vertices overlaps itself

Since I'm not familiar with a lot of mathematical terminology, I will explain this problem with a little story. Imagine you and your friend Anne have a piece of string each, and place it on a ...
3
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1answer
398 views

$2D$ Line Segment - Triangle Intersection

I've seen similar questions but could not solve my problem with those. My question is how to detect an intersection of a line segment and a triangle on a 2D coordinate system? I don't need the point ...
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0answers
34 views

“A reflection across one line in the plane is, geometrically, just like a reflection across any other line.”How?

How can this statement be represented geometrically?-"A reflection across one line in the plane is, geometrically, just like a reflection across any other line." (i tried it by drawing some ...
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1answer
1k views

What is the number of intersections of diagonals in a convex equilateral polygon?

Question: [See here for definitions]. Consider an arbitrary convex equilateral polygon with $n$-vertexes ($n\geq 4$) and the $n$-sequence $\langle \alpha_i~|~i<n\rangle$ of its angles which $\...
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2answers
532 views

deriving formula for reflection over y=mx+b using dot product

So, I know that the formula for a generic point is $$\left(\frac{1-m^2}{1+m^2}x + \frac{2m}{1+m^2}(y-b), \left(\frac{2m}{1+m^2}\right)x - \left(\frac{1-m^2}{1+m^2}\right)(y-b)+b\right)$$ when you ...
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0answers
12 views

Calculate Ellipse From 5 Points

How can I find a general or parametric form of equation for the ellipse having 5 points that lie within that ellipse? I have found this solution: Calculate Ellipse From Points?, where unfortunately ...
4
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3answers
140 views

Position of Object Suspended on a String (Need Another Answer)

I'm going to try to make as few errors in typing this as possible, so please bear with me and ask me to clarify/correct whatever needed. Q: If an object is suspended on a string hung between two ...
3
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2answers
454 views

Barycentric coordinates of a triangle

I have to do what described in the picture below. Consider the planar triangle $[p_1,p_2,p_3]$ with vertices $p_1=\begin{pmatrix}-2\\-1\end{pmatrix}$, $p_2=\begin{pmatrix}3\\-1\end{pmatrix}$...
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2answers
58 views

For $a>b>c>0$,the distance between $(1,1)$ and the point of intersection of the lines $ax+by+c=0$ and the $bx+ay+c=0$ is less than $2\sqrt2$

For $a>b>c>0$,the distance between $(1,1)$ and the point of intersection of the lines $ax+by+c=0$ and the $bx+ay+c=0$ is less than $2\sqrt2$,then $(A)a+b-c>0$ $(B)a-b+c<0$ $(C)a-b+c>...
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0answers
24 views

Affine transformations of the plane

Please help me to find the common form of affine transformations of the space $\mathbb{R}^3$ that transform the given plane $Ax + By + Cz + D = 0$ to itself. That is, all the points of this plane have ...
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10answers
117 views

Find the value of $h$ if $x^2 + y^2 = h$

Consider equation $x^2 + y^2 = h$ that touches the line $y=3x+2$ at some point $P$. Find the value of $h$ I know that $x^2 + y^2 = h$ is a circle with radius $\sqrt{h}$. Also, since $y = 3x + 2 $ ...
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0answers
36 views

Calculus & Analytic Geometry VS Vector Calculus

This question may be applicable for Academia SE, however this is strictly math-oriented and requires math whizzes' opinions. I intend to go to a tech institute to get a BS majoring in Computer ...
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1answer
94 views

Can the boy escape the teacher for a regular $n$-gon?

This is related to Prove that the boy cannot escape the teacher Suppose there is a boy in the center of a regular $n$-gon. The teacher is on the edge of the $n$-gon (but cannot leave the edge) and ...
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1answer
22 views

Given points P (2,3),Q (4,-2),R (a,0) what should be the value of a if |PR-RQ| Is maximum?

Given points P (2,3),Q (4,-2),R (a,0) what should be the value of a if |PR-RQ| Is maximum ? I tried that maybe the points are collinear but I'm getting wrong answer applying collinearity condition i....
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1answer
24 views

Vector questions about finding magnitudes, dot products, and angles.

I am given the following problem: Let $\Vert \overrightarrow{a}\Vert = 3$ , $\Vert \overrightarrow{b}\Vert = 2$ and $\angle \left(\overrightarrow{a},\overrightarrow{b}\right) = 60^\circ$. Find $\...
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0answers
36 views

General equation of a cone

What is the general equation of a cone in $\mathbb{R}^3$ space? There should be no assumptions about the location of the vertex, direction of the axis or aperture angle, these should all be variable.
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0answers
24 views

Small circles on sphere: finding angles for constant “cosine” onto a parallel.

My problem can be best explained starting from a 2D example: Imagine having a circle and wanting to discretize N points on the circumference of the circle so that the difference of the cosine of each ...
2
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1answer
38 views

Is the flow of an analytic vector field also analytic?

Let $X$ be an analytic vector field on a smooth manifold. Is it true that the flow $\Phi_t:M\to M$ associated to that vector field is also analytic?
2
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2answers
64 views

Prove that the equation of the cone $yz(\frac{b}{c})+zx(\frac{c}{a}+\frac{a}{c})+xy(\frac{a}{b}+\frac{b}{a})=0$

The plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ cuts the coordinate axes in $A,B,C.$Prove that the lines passing through the origin and intersecting the circle $ABC$ generate the cone $yz(\frac{b}{c}...
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2answers
41 views

Finding an equation of a circle

My math homework are finding an equation of a circle. Given that the center is at (-10,0) and passes through A(-6,3). Second item is the given center is at (-4, 6) and is tangent to the axis. I've ...
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1answer
75 views

Show that the vertex lies on the surface $z^2(\frac{x}{a}+\frac{y}{b})=4(x^2+y^2)$

Two cones with a common vertex pass through the curves $z^2=4ax,y=0$ and $z^2=4by,x=0.$ The plane $z=0$ meets them in two conics which intersect in four concyclic points.Show that the vertex lies on ...
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1answer
681 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
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0answers
25 views

Find the plane which touches the cone $x^2+2y^2-3z^2+2yz-5zx+3xy=0$ along the generator whose direction ratios are $1,1,1.$

Find the plane which touches the cone $x^2+2y^2-3z^2+2yz-5zx+3xy=0$ along the generator whose direction ratios are $1,1,1.$ Let the plane touches the cone at $(\alpha,\beta,\gamma)$. We know that ...
1
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1answer
39 views

Find function for graph

I would like to find a function for the following graph: I have drawn the graph myself, so not all subtle bends are to be replicated. I have noted the important points the graph should have in the ...
1
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1answer
48 views

Given: 2 lines containing the diameter of a circle and a point lying on this circle; Find: the equation of this circle

The lines $ y = \frac{4}{3}x - \frac{5}{3} $ and $ y = \frac{-4}{3}x - \frac{13}{3} $ each contain diameters of a circle. and the point $ (-5, 0) $ is also on that circle. Find the equation of this ...
3
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3answers
4k views

Find the coordinate of third point of equilateral triangle.

I have two points A and B whose coordinates are $(3,4)$ and $(-2,3)$ The third point is C. We need to calculate its coordinates. I think there will be two possible answers, as the point C could be on ...
4
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1answer
83 views

Mathematical description on the interface of two adjacent bodies.

I am recently studying about a problem related to shortest path. I can briefly describe my idea but I am not sure if there is some "professional" mathematical description about it. In the following ...
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2answers
34 views

How to find the equation for the circle when…

A circle goes trough two points, $A=(-1,2)$ and $B=(3,0$). You also know that the center of the circle is an element of the following linear equation: $$k \leftrightarrow 2x+y+3=0 .$$ How can you ...
35
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9answers
15k views

Is there an equation to describe regular polygons?

For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides ...
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1answer
14 views

Planes through $OX$ and $OY$ include an angle $\alpha,$ show that their line of intersection lies on the cone $z^2(x^2+y^2+z^2)=x^2y^2\tan^2\alpha$

Planes through $OX$ and $OY$ include an angle $\alpha,$ show that their line of intersection lies on the cone $z^2(x^2+y^2+z^2)=x^2y^2\tan^2\alpha$ The lines of intersection of the planes through $...
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1answer
71 views

Locus of the center of the circle of radius $a$,which always intersects coordinate axes

If the axes are rectangular, show that the locus of the center of the circle of radius $a$,which always intersects coordinate axes is $x\sqrt{a^2-y^2-z^2}+y\sqrt{a^2-z^2-x^2}+z\sqrt{a^2-x^2-y^2}=a^2$ ...
1
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1answer
54 views

Longest distance to the foci or the center that a point within the ellipse can be?

Given an ellipse $E$ (with the foci $f_1$ and $f_2$ and the center $c$), and a point $p$, which is the maximum distance that $p$ can be to all these 3 points to be within the ellipse $E$? I.e., which ...
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2answers
28 views

Two points are given as A(2,0) and B(8,0). What's the value of y (y>0), so that C(0,y) is such that angle ACB has maximum value?

My first guess is that it could be found as first derivative of some function, but I don't have idea what that function could be.
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2answers
202 views

How to calculate volume of a right circular cone's hyperbola segment?

PROBLEM I am working on calculating volumes of geometric solids. All shapes have been pretty basic until now. I am bewildered on how to attack the problem of calculating the volume of a slice of a ...
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4answers
41 views

What are the coordinates of the intersection points of two circles?

You have 2 circles that intersect in 2 points. You know the coordinates of their centers and you also know their radius. My question is: What are the coordinates of these 2 intersection points?
3
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2answers
42 views

$2$ points on a curve have a common tangent

Let $2$ points $(x_1,y_1)$ and $(x_2,y_2)$ on the curve $y=x^4-2x^2-x$ have a common tangent line. Find the value of $|x_1|+|x_2|+|y_1|+|y_2|$. It seems to me that I a missing a link and hence the ...
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0answers
20 views

Lorentzian Peak for Ellipse

I have the $(h,k)$ center coordinates, semi-major axis and semi-minor axis of an ellipse. I also have the height of the 2D Lorentzian peak, which is equivalent to the height of all the 1D Lorentzians (...
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0answers
8 views

Show that the $ZX-$ plane cuts it in the curve $F(\frac{bx}{x-a},\frac{cx-az}{x-a})=0,y=0.$

The vertex of the cone is $(a,b,c)$ and $YZ$-plane cuts it in the curve $F(y,z)=0,x=0$.Show that the $ZX-$ plane cuts it in the curve $F(\frac{bx}{x-a},\frac{cx-az}{x-a})=0,y=0.$ Let the equation ...
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3answers
37 views

Distance of closest aproach [closed]

A particle is kept at rest at origin. Another particle starts from $(5,0)$ with a velocity of $-4i+3j$. Find the closest distance of approach.
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0answers
17 views

The section of a cone whose vertex is $P$ and guiding curve $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,z=0$ by the plane $x=0$ is rectangular hyperbola.

The section of a cone whose vertex is $P$ and guiding curve the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,z=0$ by the plane $x=0$ is rectangular hyperbola.Show that the locus of $P$ is $\frac{x^2}{a^...
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0answers
13 views

Oblate Spheroidal Coordinates, Confocal Ellipsoidal Coordinates and Geodesy

What is the name of the orthogonal coordinate system that is most commonly used in modern geodesy\geomatics engineering to model the reference ellipsoid? I suspect it is either oblate spheroidal ...