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

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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
15 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
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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
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
23 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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38 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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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 ...
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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 ...
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1answer
42 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 $...
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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.
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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}...
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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 ...
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2answers
44 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
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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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
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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 ...
2
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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 ...
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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'...
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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 ...
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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^...
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2answers
41 views

Lines tangent to two circles

I'm trying to find the lines tangent to two circles. I've seen several examples but with poorlyy explained methods. Given the circle $(x-x_{0})^2+(y-y_{0})^2=r_{1}^2$ and the the line equation $y=...
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1answer
26 views

How to find the coordinates of the points $ T$ and $T'$

Referring to the accompanying figure,how to find the coordinates of the points $T$ and $T'$, where the lines $L$ and $L'$ are tangent to the circle of radius $1$ with center at the origin.
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1answer
52 views

Show that pair of straight lines $ax^{2}+2hxy+ay^{2}+2gx+2fy+c=0$… meet coordinate axes in concyclic points.Also find equation of

Show that pair of straight lines $ax^{2}+2hxy+ay^{2}+2gx+2fy+c=0$ meet coordinate axes in concyclic points. Also find equation of the circle through those cyclic points My Attempt Given equation ...
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2answers
126 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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1answer
32 views

Points on the curve

We have to find points on the curve $ax^2+ay^2+2 bxy=c$ (Where c>b>a ) whose distance from origin is minimum . I am not getting any start . I am able to just find that the curve would be hyperbola
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2answers
74 views

Equation of common tangent.

What is the equation of common tangent to the circle $(x-3)^2+y^2=9$ and parabola $y^2=4x$.$$My Try$$ So equation of tangent at point $(x_1,y_1)$ is $xx_1+yy_1-3(x+x_1)=0,yy_1=2(x+x_1)$ for circle,...
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2answers
155 views

A very little known approximation for the lesser angle of a triangle

In the article A note on an Approximation in Trigonometry is proved a very interesting approximattion to the lesser angle of a triangle (in degrees): $ (1) A \approx \frac{344\Delta}{2s(s-a)+bc}$ ...
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1answer
24 views

Length of normal. chord…

What is the length of normal chord which subtends right angle at the vertex of parabola $y^2=4x$. $$My Try$$ let the equation of normal be $y=mx-am^3-2am$ Now I assumed slope of this line as $45$ (...
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0answers
26 views

Can I define a 2-D ellipse via two points and their respective tangents? [duplicate]

I have encountered a rather complex issue that I could not solve on my own, and my research online has so far not yielded a suitable answer to my question, hence my decision to seek help in this place....
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1answer
37 views

Length of a chord parallel to the minor axis at a distance $d$ on a rotated ellipse

In this old question an equation was posted for something similar: Equation for the length of a chord parallel to either the minor or major axis in an ellipse Anybody knows from where this equation ...
4
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2answers
119 views

Nature and number of solutions to $xy=x+y$

Find all solutions to $$xy=x+y$$ Initially the given condition was $x,y\in \Bbb{Z}$. $$$$In this case, I just guessed that the solutions were $(0,0)$ and $(2,2)$. As far as I can see, these are the ...
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0answers
10 views

What is the way to derive the equation of director sphere of any central conicoid?

I am following a book on analytical solid geometry. The book defines the director sphere of a central conicoid as the locus of a point which lies on the intersection of three mutually perpendicular ...