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

learn more… | top users | synonyms (1)

1
vote
2answers
29 views

Finding the intersection of an xy-plane in a 3D-Coordinate System

I found the equation of a sphere that has a center of $(1,-12,8)$ with a radius of 10 and I got the following equation: $(x-1)^2 + (y+12)^2 + (z-8)^2 = 100$ As for finding an intersection for the ...
0
votes
1answer
29 views

Area of a triangle - straight lines

Question: What is the area of the triangle formed by the line $x + y = 3$ and angle bisectors of the pair of straight lines $x^2 - y^2 + 2y = 1$. Well I really have no idea how to even start the ...
5
votes
1answer
51 views

Shortest path between two points via two disks

Hallo everybody, I have the following problem regarding shortest paths in $R^2$. Suppose you are given two points $p$ and $q$ and two unit disks, as in the picture. I am looking for a path from ...
0
votes
0answers
30 views

to prove the following two lines are parallel [on hold]

I have to prove that the following two lines are parallel. Your help would be much appreciated. $r=2i+3j+t(i-4j)$ and $r.(8i+2j)=5$
0
votes
1answer
34 views

Show that $f(x)$ satisfy the differential equation

Given a curve $C=\{(x,f(x)\in \mathbb{R}\times\mathbb{R}\mid x\in(r_1,r_2)\}$ with has the following property.(f(x) is $C^3$-function) At any point $(a,f(a))\in C$ if we change coordinate system by ...
0
votes
0answers
8 views

Extract equations of dependency between two projected views

Regarding question Finding equation of an ellipsoid, the answer says that we have the following equation between projections on XY & XZ plane: $$\frac{Z_3^2}{Z_2} - Z_1 = \frac{Y_2^2}{Y_3} - Y_1$$ ...
1
vote
0answers
31 views

How to solve the sets of equations to find the matrix of coefficients for an ellipsoid

Regarding the below question: Finding equation of an ellipsoid I have two more questions: 1- In the "update" section of answer provided by @achillehui I can not understand the method he described for ...
-3
votes
0answers
41 views

How to show that if a complex function is analytic then it is infinitely many times differentiable geometrically? [duplicate]

I am going through the theorem which proves that if a complex valued function is analytic than it is infinitely many times differentiable. But I am not sure how to explain this geometrically without ...
2
votes
3answers
60 views

Finding the equation of a line whose segment is intercepted between axes

The question is: Find the equation of a line through (-2, 5) and whose segment intercepted between axes in the 2nd quadrant is 7√2 I have two graphs in mind but I don't know which one is correct. The ...
0
votes
2answers
20 views

proves of parametric curves via parametric equations

Hi could anyone help me with this problem. An astroid is given by the equation $$x^{2/3} + y^{2/3} = 1.$$ Prove via parametric equations that the length of a piece of a tangent line between the ...
0
votes
0answers
24 views

Holomorphic and meromorphic functions on Riemann surfaces

On any domain $\Omega\subset \mathbb{C}$, the set of all holomorphic functions form an integral domain. Its field of quotient is the set of all meromorphic functions on $\Omega$. However this is not ...
1
vote
1answer
16 views

Vectors in 3 dimensions

If $a$ is a vector that makes equal angles with ${\mathbf i},{\mathbf j},{\mathbf k}$ and has magnitude $3$, then find the angle of $a$ with either of these unit vectors? Wouldn't the answer simply ...
5
votes
2answers
48 views

How is Cartesian coordinate system related to his philosophy

In 1637, Rene Descartes published his famous monograph about philosophy "Discourse on the Method of reasoning well and Seeking Truth in the Sciences", and analytic method of geometry has been come up ...
1
vote
2answers
134 views

Finding equation of an ellipsoid

Consider I have an ellipsoid (let say an egg) lies in a general form in 3D space. Suppose, I have the equations of two projected views of this egg (e.g. one projected view on x-y plane and another one ...
1
vote
1answer
30 views

How to find the angle between two vectors?

Here, I would like to describe my requirements .. Let's say we have two vectors named $\bf A$ and $\bf B$. Two vectors are in different magnitude and opposite directions and lay on different planes. ...
0
votes
2answers
36 views

How to show that a given line has a certain equation?

Say line $A(3,0)$ and $B(0,2)$ How do I 'show' that they have equation $2x + 3y - 6 = 0$?
1
vote
0answers
39 views

Questions about circle

I found the following problem from a book. Let A = (-1, 0), B = (1, 0) and k = a constant which is not equal to 1. C(x, y) is a variable point such that AC = kBC. Find the locus of C. The ...
0
votes
1answer
79 views

How is the curve with equation $1/x^4 + 1/y^4 = 1$ called?

Well what is the graph for $$\frac 1{x^4} + \frac 1{y^4} = 1$$ called? According to $ Wolfram-Alpha$: http://www.wolframalpha.com/input/?i=plot+1%2Fx%5E4%2B1%2Fy%5E4%3D1+and+y%3Dx+and+y%3D-x ( ...
2
votes
3answers
48 views

Pair of straight lines

Question: Find the equation of the bisector of the obtuse angle between the lines $x - 2y + 4 = 0$ and $4x - 3y + 2 = 0$. I don't even know how to proceed here. I know how to find the angle between ...
-1
votes
0answers
36 views

Ellipsoid-Sphere Intersection [duplicate]

Suppose I have an Ellipsoid given as: $$\frac{(x-x_2)^2}{a^2} + \frac{(y-y_2)^2}{b^2} + \frac{(z-z_2)^2}{c^2} = 1$$ And a Sphere $$(x-x_1)^2 + (y-y_1)^2 + (z-z_1)^2 = R_1^2$$ If they intersect ...
3
votes
1answer
17 views

If $P=(x_0,y_0)$ is a point in a focal chord of the parabola $x^2=4py$ then find the coordinates of the other point

$\textbf{Exercise:}$ If $\overline{PQ}$ is a focal chord of the parabola $x^2=4py$ and the coordinates of $P$ are $(x_0,y_0)$, show that the coordinates of $Q$ are $$ ...
1
vote
1answer
40 views

Show an equation of a line passing through $P$ and parallel to the line given by $ax+by+c=0$.

Question: A person considers lines on the plane $\mathbb{R^2}$ to be solutions of equations of the form $ax+by+c=0$, where $a,$ $b,$ and $c$ are fixed reals satisfying $a^2+b^2\neq0$. Give a point ...
1
vote
2answers
24 views

How to define a cloud of points relative to a vector path?

I've been researching and playing with examples of particle clouds in a graphics visualization. Most use shape geometries to define a field of particles, or parameters for distributing them randomly ...
3
votes
0answers
63 views

My orbiting body is orbiting about the wrong focus of it's elliptical orbit… why? [closed]

I am coding in c++ and am computing the position of an orbiting body as a function of time. Everything is almost working. I have a nice elliptical orbit. Except, my orbiting body speeds up as it ...
1
vote
0answers
20 views

Equation of a line through a point and another line

I need to get the equation of a line that passes through the point Q(6, 3, 2) and intersects: $$L: (1, -1, 4) + t(0, -1, 1)$$ and forms an angle of 60° What I did so far: The direction vector of L ...
3
votes
2answers
26 views

Parallel plane that contain lines

I have the lines: $$L_1: \frac{x-3}{2}= \frac{y+5}{-3} = \frac{z+1}{5} ,$$ $$L_2: \frac{x+1}{-4}=\frac{y-1}{3}=\frac{z-3}{-1}$$ I need the equations of the parallel planes $P_1$ and $P_2$ that ...
0
votes
1answer
62 views

Finding Shortest distance between a Sphere and Ellipsoid?

Suppose that ,I have a Sphere and an ellipsoid as Sphere: $(x-x_1)^2 + (y-y_1)^2 + (z-z_1)^2 = R_1^2$ Ellipsoid: $\large\frac{(x-x_2)^2}{a^2} + \frac{(y-y_2)^2}{b^2} + \frac{(z-z_2)^2}{c^2} = 1$ ...
-2
votes
1answer
19 views

Computing vector products [closed]

Can someone help me with these calculations? i) $(3\mathbf{i}-2\mathbf{j+k}) \times (\mathbf{i}-4\mathbf{j}+3\mathbf{k})$ and ii) $(2\mathbf{i}-3\mathbf{j}-\mathbf{k})\times ...
2
votes
3answers
107 views

What is the cone of the conic section?

Given the general (real valued) equation of a conic section: $$ A x^2 + B xy + C y^2 + D x + E y + F = 0 $$ Then what is the circular cone associated with it ? Is it unique ? And is there a way to ...
2
votes
2answers
38 views

Analytic Geometry: One sheeted hyperboloid

Good afternoon! I have a question about analytic geometry. I don't actually know if the answer is quite simple, and I missed something while revising, or if it is actually more complicated than I ...
2
votes
1answer
28 views

Determination of a volume.

Consider, in the Cartesian plane, the square Q having vertices in the points $(-1, 0), (1, 0), (0, -1)$, and $(0, 1)$. The sections of a solid with planes orthogonal to $y=0$ are squares having two ...
1
vote
2answers
39 views

Rotation around a line which is determined by two points in 3D space

If we have three points like $A(x_1,y_1,z_1)$, $B(x_2,y_2,z_2)$ and $C(a,b,c)$. Then, $A$ and $B$ determines a line like $l$. After that, we rotate $C$ around $l$ by $\omega$ degree (anti-clockwise). ...
0
votes
0answers
40 views

A book on analytic geometry

It's easy to find good recommendation for books here for any subject other than analytic geometry ,therefore I'd like to ask for any suggestion of analytic geometry books ,the only charactrestic I'm ...
2
votes
0answers
56 views

A variational strategy for finding a family of curves?

In a recent question, I asked for examples of families of distinct smooth curves with fixed area and perimeter (which for this question I will dub as doubly-isometric). That wording allows $C^1$ ...
0
votes
3answers
69 views

Equation of rectangle

I need equation of a rectangle on the Cartesian coordinate system. Is there an equation for a rectangle? for example equation of ellipse is $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
0
votes
1answer
30 views

Given that the graph of $f$ passes through the point $(1, 6)$ and that the slope of its tangent line at $(x, f(x))$ is $2x + 1$, find $f(2)$.

As in the title - we assume that the graph of $f$ passes through $(1,6)$ (i.e. $f(1) = 6$) and that the slope of its tangent line at $(x, f(x))$ is $2x + 1$ and we are asked to find $f(2)$. How does ...
1
vote
1answer
23 views

What is the basic idea of homogenisation of an equation?

I do get that when you are homogenising it makes it in an equation of pair of straight lines passing through origin but what is its actual point and its applications?
1
vote
1answer
35 views

Lines joining origin to points of intersection of two conics

If the lines joining origin and point of intersection of curves $$ax^2+2hxy+by^2+2gx=0$$ and $$a_1x^2+2h_1xy+b_1y^2+2g_1x=0$$ are mutually perpendicular, then prove that $$g(a_1+b_1)=g_1(a+b)$$ How ...
0
votes
1answer
23 views

Coordinate Geometry of circles; Radical Axis question

If one of the diameters of the circle $x^2+y^2-2x-6y+6=0$ is a chord to the circle with center at $(2, 1)$, then the radius of the second circle is? Apparently the solution is $3$, with the ...
2
votes
1answer
53 views

Help with simple rotation on an x,y plane

I'm a programmer, with too little background in mathematics, and I am currently faced with the challenge of rotating an object on a 2 axis plane. Something that is hopefully quite easy for you guys. ...
2
votes
2answers
36 views

Calculus III: Find the points of the curve…

I have to find the points of the curve $$r\left( t \right) =\left( t,{ t }^{ 2 },{ t }^{ 3 } \right) $$ where the osculating plane passes through the point $\left( 2,-\frac { 1 }{ 3 } ,-6 \right)$.
0
votes
1answer
18 views

Coordinates rotation and function change

In the Cartesian coordinates $(x,y)$, I have a vector function $\bar{f}(x)=\hat{x}A\cos(yk)$, where $A$ and $k$ are constants. I make now a 45 degrees rotation (in the same plane) to the new set of ...
0
votes
1answer
28 views

Three planar vectors $x,y,z$ such that $x$ is orthogonal to $y + z$ and $z$

Let $x$ be a non-zero vector, orthogonal to vectors $y + z$ and $z$, with $x, y, z \in \mathbb R^2$. Prove that $y$, $y - z$ and $z - y$ are orthogonal to $x$ and parallel to $z$. To prove they ...
1
vote
0answers
47 views

intersection of a line and plane on a 3-sphere

Suppose I have two 4D points, $\mathbf{a}=(a_1,a_2,a_3,a_4)$ and $\mathbf{b}=(b_1,b_2,b_3,b_4)$, that both lie on a unit 3-sphere (i.e. unit distance from origin). In addition, I have a 2-D plane that ...
9
votes
7answers
417 views

Constructing a family of distinct curves with identical area and perimeter

Two recent questions were posed by Arjuba [1] [2] asking for counter-examples regarding whether two different figures could have the same perimeter and area. Responders quickly raised a number of such ...
1
vote
3answers
37 views

Doubts on locus and its equation

"Find the equation to the locus of a point which is collinear with points $M(a,0)$ and $N(0,b)$." The answer is $- x/a + y/b$. How I tried to find the solution: $P$ is a point whose assigned ...
0
votes
2answers
53 views

Determine the matrix of the reflections in the fol­lowing plane in $\Bbb R^3$. [closed]

$2x_1-2x_2-x_3= 0$ How would I go about approaching this problem?
2
votes
2answers
59 views

Parametric formula for figure 8 mobius strip

I'm making 3D prints with Mathematica, and am interested in a parametric formula for a mobius strip that is in the form of a figure 8, rather than simply a circle with a twist in it. Can someone help ...
0
votes
1answer
32 views

Equation for the length of a chord parallel to either the minor or major axis in an ellipse

I am looking for a way to compute the length of any chord parallel to the minor (or major) axis of an ellipse. In all cases I know the lengths of both axes, and the distance between the chord and axis ...
0
votes
0answers
5 views

Find gap coordinates to connectable reference Rectangle

Sorry for my english. This is my first question. I need to find gap coordinates of items to reference object programmaticly. Item positions can be changed. How can i do this with an efficent way? ...