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

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4answers
100 views

Graph of the function $\cos(x)\cos(x+2)-\cos^2(x+1)$ will be?

Graph of the function $\cos(x)\cos(x+2)-\cos^2(x+1)$ will be? (A)A straight line (B)A parabola Give the corresponding equation too. Source:JEE 1997. Can ...
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0answers
38 views

Equations of the tangent planes to the sphere

Find the equations of the tangent planes to the sphere $x^2+y^2+z^2-10x+2y+26z-113=0$ which are parallel to the straight lines $\frac{x+5}{2}=\frac{y-1}{-3}=\frac{z+13}{2}$ and $\frac{x+7}{3}=\frac{y+...
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1answer
17 views

2D AABB count vector for not having collision after movement

I hope it is correct here, I feel like this question is more math related than programming. Table of Contents Introduction What is question about / problem description What way I figured out Other ...
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0answers
13 views

Condition for n points in the plane to determine a convex n gon

Suppose there are n points in the plane, labelled 1 through n, no three of which lie on a line. Suppose further that for every triple [i,j,k] with i< j < k that travelling from i to j to k is ...
2
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2answers
44 views

Questions regarding dot product (possible textbook mistake)

I am given the following exercise: Show that $\Vert \overrightarrow{a} + \overrightarrow{b} \Vert = \Vert \overrightarrow{a} \Vert + \Vert \overrightarrow{b} \Vert $ if and only if $\...
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1answer
45 views

Necessary & Sufficient condition for the line $ax+by+c=0$ to pass through the 1st quadrant

What is the necessary and sufficient condition for the line $ax+by+c=0$, where $a,b,c$ are non-zero real numbers, to pass through the first quadrant? I could find the points at which the line ...
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0answers
44 views

The locus of the orthocentre of the triangle formed by the lines

The following question is from IITJEE 2009 paper. The locus of the orthocentre of the triangle formed by the lines $(1 + p)x − py + p(1 + p) = 0$ $(1 + q)x − qy + q(q + 1) = 0$ and $y = 0$, where $...
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0answers
15 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, ...
0
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1answer
22 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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4answers
36 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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0answers
11 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
76 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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1answer
62 views

Ellipse and parallel 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 parallel lines - ...
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1answer
10 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 ...
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0answers
37 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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0answers
16 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 ...
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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 ...
2
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10answers
119 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
42 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
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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1answer
24 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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0answers
26 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 ...
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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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0answers
39 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
28 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 ...
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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
54 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 ...
2
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1answer
43 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?
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3answers
143 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 ...
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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 ...
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1answer
15 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 $...
1
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1answer
79 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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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.
2
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2answers
67 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}...
1
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1answer
60 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 ...
3
votes
2answers
45 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
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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0answers
19 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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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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4answers
42 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?
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0answers
15 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 ...
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1answer
16 views

Projection of vectors over along their axis

I have difficulties to understand first how does the strong blue vectors appear Second, how does the light blue vector $w=u\times v$ appears? I thought it was going to be $\vec 0$
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2answers
14 views

Projection of a point on a line

Find the projection of the point $(-6,4)$ onto the line $4x-5y+3=0$ I can find the distance between the point and the line, but I do not think it can help
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0answers
36 views

Properties of polyhedron solving constrained max problem

This is a question for people who don't have trouble to think in more than two dimensions. Don't hesitate to ask clarifying questions! Let us suppose we have $n$ random variables $X_i$ that are iid ...
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3answers
450 views

Why are there two versions of a polar equation for a circle from geometric form

In class today we learned that a rectangular/geometric equation for a circle such as $x^2+(y-5)^2 = 9$ can be converted into a polar equation by reducing it to the quadratic equation $r^2-10r\sin \...
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1answer
88 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 ...
3
votes
1answer
37 views

finding a function given a slope and a point

I need to find the function $f(x)$ that is tangent to a line whose slope is given by $\displaystyle \frac{(1+\sqrt x)^{\frac{1}{2}}}{8\sqrt x}$ that passes through the point $(9,8/9)$. I really don'...
2
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0answers
44 views

Find the ratio of slope

Note : Elevation $46000$ and all dimention in $mm$ (milimeter) The pipe will be installed on a surface of module structure, that module structure has different surface. I want to know " ratio ...
1
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1answer
33 views

Find the range of a

If the point $P(a^2,a)$ lies in the region corresponding to the acute angles between the lines $2y=x$ and $4y=x$ then range of a is... This would have been easy if the lines had constants and I ...
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1answer
22 views

A straight line moves so as to meet the straight lines

A straight line moves so as to meet the straight lines $y=mx, z=c$ and $y=-mx, z=-c$ in A and B and intersects the curve $yz=k^2, x=0$, show that the locus of the middle point of $AB$ is $$(m^...