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

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0
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1answer
290 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 ...
0
votes
1answer
25 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 ...
1
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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
2answers
27 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 ...
5
votes
1answer
49 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 ...
1
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4answers
349 views

Area of a quadrilateral

The perpendicular bisector of the line joining $A(0,1)$ and $C(-4,7)$ intersects the $x$-axis at $B$ and the $y$-axis at $D$. Find the area of the quadrilateral. Thank you in advance!
1
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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 ...
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 ...
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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: $$\frac{Z_3^2}{Z_2} - Z_1 = \frac{Y_2^2}{Y_3} - Y_1$$ ...
-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
1answer
227 views

How to show that a line pass through a point?

How to show that a line pass through a point? Two fixed straight line $OX$ and $OY$ are cut by a variable line at the points $A$ and $B$ respectively and $P$ and $Q$ are the feet of the ...
1
vote
2answers
85 views

Perpendicular form of the straight line equation.

There are 5 to 6 standard forms of the straight line equation. for example slope intercept form, two intercept form, point slope form and perpendicular form. I have clear visualization of all forms ...
2
votes
2answers
115 views

An equation for all those points that have the same shortest distance to the same straight line in 3D space.

Can you form an equation for a ''pipe'' in 3D space? It means all those points P(x,y,z) that have the same shortest distance for the same straight line l. For example what would the pipe equation be ...
1
vote
1answer
46 views

shortest distance between two cones in 3-dim space

How can I find the shortest distance between two cones in 3-dim space? cone 1: apex - $(x_{0}, y_{0}, z_{0})$ angle - $\alpha_{0}$ base circle - $(cx_{0}, cy_{0}, cz_{0}, r_{0})$ cone 2: apex - ...
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 ...
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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 ...
1
vote
1answer
562 views

Find equations of the ellipses given conditions on the directrices, foci, and vertices

The ellipses have their centers at the origin and their major axes on the $x$-axis. Find the equation: with distance between directrices $27$, and between foci $3$; with a focus at $(-\sqrt{13},0)$ ...
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 ...
-1
votes
2answers
89 views

Is it possible to find the coordinates of a point in 3D space, given its distance from a known point?

Is it possible to find the coordinates $(x,y,z)$ of a point in $3d$ space when given: A) the unknown point is $(x,y,z)$. B) the known point is $(a,b,c)$. C) the distance between the two points is ...
1
vote
1answer
232 views

Intersection of two lines

What is the suggested method to find the intersection of two line *segments in 3D space programmatically? I mean there are various methods to solve a set of 2 linear equations, eg. Using ...
0
votes
3answers
564 views

Find the area of the triangle using analytic geometry

We have a $\triangle ABC$ with: Base $AB$ with length 14 $AC$ with length 15 $BC$ with length 13 Find the area of the triangle using analytic geometry.
0
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1answer
1k views

Given two vertices, how to find the other two vertices of a rhombus?

$A\;(-3,-4) $ and $ C \; (5,4)$ are the ends of the diagonal of a rhombus $ABCD$. Given that the side BC has gradient $\frac{5}{3}$; How could we find the coordinates of $B$ and hence of $D$? ...
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 $2x + 3y - 6 = 0$?
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 ( ...
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 ...
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
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
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
2answers
376 views

The intersection of a line with a circle

Get the intersections of the line $y=x+2$ with the circle $x^2+y^2=10$ What I did: $y^2=10-x^2$ $y=\sqrt{10-x^2}$ or $y=-\sqrt{10-x^2}$ $ x+ 2 = y=\sqrt{10-x^2}$ If you continue, $x=-3$ or ...
1
vote
1answer
29 views

Equation of circumscribing circle

Show that a cyclic quadrilateral is formed by the lines $5x+3y=9$, $x=3y$, $2x=y$ and $x+4y+2=0$ taken in order. Find the equation of the circumscribing circle. How do i go about it?
1
vote
1answer
27 views

How to calculate new position of a rectangle after translation and rotation?

I have a rectangle - lets say 100 long by 75 high. Origin been bottom left corner. I move the rectangle up and across by 10 and rotate by 3 degrees from centre of part. How do I calculate the new ...
0
votes
2answers
88 views

A point P moves so that AP and BP are perpendicular. Find the equation of the locus of P

A point p(x,y) moves so that AP and BP are perpendicular, given A=(3,2) and B =(-4,1). Find the equation of the locus of P. Can someone please advise me on what to do for this question. Just need a ...
-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 ...
1
vote
2answers
44 views

What is a homographic solution in three body problem?

I came across Saari's homographic conjecture in Three Body problem. I need more information on what exactly is a homographic solution and how is it different from a homothetic solution?
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6answers
6k views

Horizontal tank with hemispherical ends depth to capacity calculation

I am trying to find an accurate way of calculating the capacity of an underground tank at a given depth. The tank manufacturer has provided a strapping table for the tank which tells me the capacity ...
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 ...
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
25 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
2answers
242 views

Why is the locus of the centres of the circles passing through two points is the perpendicular bisector of the two points?

Why is the locus of the centres of the circles passing through two points is the perpendicular bisector of the two points?
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 ...
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: $\large\frac{(x-x_2)^2}{a^2} + \frac{(y-y_2)^2}{b^2} + \frac{(z-z_2)^2}{c^2} = 1$ ...
0
votes
2answers
302 views

Q). Show that the four points are angular points of a rectangle$ (0,-1) (4,-3) (8,5) (4,7)$.

I started to solve the question by taking the sides of rectangle ABCD then added a midpoint in the rectangle and divided the rectangle in diagonal then found out the midpoint of diagonals AC and BD ...
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 ...
0
votes
4answers
126 views

How do I find the center and radius of this circle? [closed]

How do I find the center and radius of this circle? $$4x^2+4y^2+24x-16y+41=0$$
-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 ...