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

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2
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1answer
223 views

Metric tensor of complex numbers & Hamiltonian Mechanics

The Euclidean $\mathbb{R}^2$ geometric space can be mapped onto $\mathbb{C}$. In other words I see it like this $$\vec{v} = x\vec{x}+y\vec{y} = x\vec{1}+y\vec{i}= \begin{bmatrix}x \\y\end{bmatrix} ...
0
votes
1answer
147 views

At least two circles meeting these cond. have nonempty intersection

Here is a problem I've been trying to solve for some time now. Maybe you could help me. We have two sets $\mathcal {S}$ is a family of circles in the plane such that for any $x \in \mathbb{R}$ there ...
2
votes
1answer
65 views

Plugging in a point not on a plane, into the plane's equation

Say a plane P has a given equation ax+by+cz=d. Given a point $(x_0, y_0, z_0)$ that is not included in P. When $(x_0, y_0, z_0)$ is plugged into $f(x,y,z)=ax+by+cz-d$ and it outputs some nonzero ...
3
votes
2answers
640 views

Find the standard form of the conic section $x^2-3x+4xy+y^2+21y-15=0$

Find the standard form of the conic section $x^2-3x+4xy+y^2+21y-15=0$. I understand the approach in trying to solve these problems. But the $4xy$ is confusing me. I am not sure of where to start on ...
2
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0answers
120 views

Find $k$ such that the intersection of $x+ky=1$ and $y^2 - x^2 - z^2 = 1$ is an ellipse or a hyperbola

Find the values of $k$ such that the intersection of the plane $x+ky=1$ with the two-sheeted elliptic hyperboloid $y^2 - x^2 - z^2 = 1$ is (a) an ellipse and (b) a hyperbola. My attempt is the ...
2
votes
2answers
4k views

How can you construct as many intersections as possible with n lines?

If you have $n$ lines, it seems to be obvious that you can have at most $\frac{n^2-n}{2}$ intersections: $n = 1$: Obviously you need two lines to intersect, so the maximum number of intersections is ...
6
votes
4answers
12k views

Find intersection of two 3D lines

I have two lines $(5,5,4) (10,10,6)$ and $(5,5,5) (10,10,3)$ with same $x$, $y$ and difference in $z$ values. Please some body tell me how can I find the intersection of these lines. EDIT: By using ...
2
votes
1answer
423 views

Find a plane whose intersection line with a hyperboloid is a circle

Find a plane $\pi$ which involves x-axis and its intersection line with $$\frac{x^2}{4}+y^2-z^2=1$$ is a circle. Because the plane want to be find involves x-axis,so set as $By+Cz=0$,then I must to ...
1
vote
1answer
476 views

How to find the tangent cone of the sphere

A given sphere: $$x^2+y^2+z^2+2x-4y+4z-20=0$$ How to find the tangent cone of it ? the vertex of the cone is $(2,6,10)$ thanks very much.
1
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1answer
55 views

How to show that two planes meet a hyperboloid in circles which lie on a sphere

How to show that the planes $2x+3z=5$ and $2x-3y+7=0$ meet the hyperboloid $-x^2+3y^2+12z^2$=$75$ in circles which lie on the sphere $3$$x^2+3y^2+3z^2+4x+36z-110=0$ please help.
2
votes
2answers
637 views

Is the area of intersection of convex polygons always convex?

I am interested specifically in the intersection of triangles but I think this is true of all convex polygons am I correct? Also is the largest possible inscribed triangle of a convex polygon always ...
1
vote
1answer
771 views

How to find the intersection of the area of multiple triangles

I have a couple of questions regarding finding the intersection of triangles. I have a system of 16 projectors that all have slightly different color gamuts. The color gamuts are represented by a ...
2
votes
4answers
141 views

How do I find the points of a circle?

Say you have a center of $(5, 5)$ and a radius of $2$. If you went for each x-value in $\{3, 4, 5, 6, 7\}$, how would you find the y value? EDIT: I have this code in C# ...
2
votes
1answer
204 views

Hyperbolas on an Imaginary Graph

My first question is what this type of graph (of $x-y-i$) is called since I was unable to find any information about any such graph. Now for the real question, I used the equation $\frac{x^2}{a^2} ...
2
votes
3answers
1k views

Real life coordinate geometry problem

To conduct a sport activities, in a rectangular shaped school ground $ABCD$, lines have been drawn with chalk powder at a distance of $1$ m each. $100$ flower pots have been placed at a distance of ...
0
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2answers
210 views

How to check if a line is coinciding with another line?

I asked a question on stackoverflow about how to know if a line is coinciding with another polygon. [http://stackoverflow.com/q/13304575/1362544] The answer I got suggested checking intersection of ...
2
votes
3answers
124 views

What kind of software can be used to solves systems of equations?

For example, I have to solve the following equations: $$\left\{\begin{align*} &x^2 + y^2 + z^2 = 1\\ &Ax + By + Cz = 0 \end{align*}\right.$$ for $y$ and $z$, where $x$, $A$, $B$ and $C$ are ...
5
votes
1answer
628 views

Proper mapping theorem

My professor mentioned a proper mapping theorem after the name of Remmert which says: Let $X$ and $Y$ be complex manifolds, $f:X \to Y$ be a proper holomorphic map, and $V \subset X$ be a complex ...
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3answers
350 views

Volume of a Cone — Stuck On My Approach

I'd like to calculate the volume of a right circular cone via my way. If I have a right-triangle with base $D$ and height $H$ then its area is $\frac{1}{2}HD$. Now if we imagine rotating this shape ...
1
vote
1answer
168 views

How to draw $(A,B)\sim (C,D) \implies (A,C)\sim (B,D)$ when $A,B,C,D$ are collinears?

$(A,B)$ and $(C,D)$ are parallel vectors, in the book I'm reading, it illustrates one case for this proposition: $(A,B)\sim (C,D) \implies (A,C)\sim (B,D)$ with the following figure: And then ...
0
votes
1answer
106 views

Lines which intersect the postive half axis of x

We have to find out which lines intersect the positive half axis of $x$. According to this formula we can determine if the angle between two points $(A[x_1, y_1]$ and $B[x_2, y_2]$ ) of the line ...
2
votes
0answers
449 views

Proof of coarea formula

I want to prove the coarea formula $\operatorname{Vol}(M) = \int_M d\operatorname{Vol}_M = \int_{-\infty}^\infty \frac{1}{|\nabla f|} \operatorname{area }(f^{-1}(t)) dt$ where $f: M \rightarrow ...
1
vote
2answers
1k views

Find the equation of the parabola

We have a point $A(6,0)$ and a line $k:y=2$. Show that the equation of the parabola with a locus $A$ and a directrix $k$ has the formula: $\dfrac{1}{4}x^2-3x+8$. I had a test on analytic geometry ...
0
votes
1answer
43 views

How to explain the solution

How to write an equation for plane, which includes dots with radius-vectors $\mathbf r_{1}, \mathbf r_{2}, \mathbf r_{3}$ that do not lie on a straight line? The answer is $$ (\mathbf r, ([\mathbf ...
10
votes
2answers
2k views

The intersections of 2 circles

Lets consider the following (random) question: Find the intersections of the circles $c_1: x^2+y^2=25$ and $c_2: (x-2)^2 + (y-3)^2=9$ In order to solve this we can do $c_2-c_1$, which leaves us with ...
0
votes
1answer
279 views

Difficult volume computation inside an ellipsoid and above a plane

I am taking a Calculus course currently and am stuck on the last question of my assignment. Find the volume of the region inside the ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} ...
1
vote
3answers
234 views

Area Between Curves

The problem I am working on is, "In Exercises 17 and 18, find the area of the region by integrating (a) with respect to and x (b) with respect to y." The two functions: $g(y)=4-y^2$, and $f(y)=y-2$ ...
3
votes
2answers
646 views

Calculation of Area of right angled triangle - Apostol exercise 1.7 problem 2

"Prove that every right triangular region is measurable because it can be obtained as the intersection of two rectangles. Prove that every triangular region is measurable and its area is one half the ...
0
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1answer
4k views

Distance minimum distance between point and sphere

How can I find minimum distance between point and sphere ? sphere properties : position of center a,b,c redius of the sphere R point properties position x,y,z
2
votes
1answer
497 views

Distance between cone and point

How can I find minimum distance between cone and a point ? Cone properties : position - $(0,0,z)$ radius - $R$ height - $h$ Point properties: position - $(0,0,z_1)$
1
vote
2answers
321 views

Parametric Equation Problem

The problem is, "to determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? Explain." (a) $x=t;\quad ...
1
vote
1answer
282 views

Restriction Of Parametric Functions Domain

The problem I am working on is, "Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the ...
2
votes
1answer
254 views

Sketching A Plane Curve

The problem I am working on is, "Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the ...
0
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2answers
172 views

Faster Distance formulae for higher n dimension

I need to calculate the distance two points, but these two points contain more than 100 dimensions. With the regular two or three dimension distance formula, we can extend this further to n dimension ...
3
votes
2answers
70 views

Curve of Equal SWR

I'm trying to figure out how radio frequency "matching stubs" work. In order to fully understand the problem, I need to know how the "curve of equal SWR" looks like. I did a few plots, and it looks ...
20
votes
1answer
644 views

Intuition why the volume and surface area of the unit sphere eventually decrease

The volume formula for a unit sphere, $$\frac{\pi^{n/2}}{\Gamma{(1 + n/2)}},$$ and the surface area formula, $$\frac{2\pi^{n/2}}{\Gamma{(n/2)}},$$ both attain maximum values for finite $n$. We can ...
0
votes
1answer
988 views

Help: Find the area of the shaded region

Given an arc PQ with curvature $\frac{1}{9}$ Three identical circles with radii 3 and centered at B,G,A respectively. The circumference of the circles pass through each other's centers. Find the area ...
1
vote
1answer
92 views

Vector VS Plane intersection

Could You help me with task: From point $M(3,5)$ that belongs to plane: $A(0,0), B(0,10), C(20,10), D(20,0)$, comes out vector $V$ at an angle a(with $OX$). Need to find point $X(x,y)$ at which he ...
2
votes
1answer
167 views

Chain of Circles

A chain of four circles centered at A, B, C, and D are touched on one side by the line GH and on the other side by a circular arc EF centered at O. Find the area of D in terms of the areas of A, B, ...
3
votes
2answers
516 views

How to derive this angle (about hyperbola)?

In section "Quadratic equation" of Hyperbola of Wikipedia( http://en.wikipedia.org/wiki/Hyperbola#Quadratic_equation), it it said that "The principal axes of the hyperbola make an angle Φ with the ...
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votes
1answer
1k views

analytic geometry … 2 problems

1st problem : find the equation of the straight line having slope $m$ passing through the point $(a, 0)$. What are the coordinates of the point of intersection of this line with the y-axis? 2nd ...
2
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4answers
2k views

Equation for distance from a point outside a sphere to any point on its surface

I have a point m outside a sphere. The sphere center is o and r is the radius of sphere. Distance from point m to o is l. If we draw a line from m to any point on the surface of sphere, this line has ...
2
votes
2answers
2k views

Find the area of a triangle using analytic geometry

Given are the points $P (1,0)$ and $Q (3,2)$. The points $P$ and $Q$ have the same distance to a certain line $l$, which intersects the positive x-axis in the point $A$ and the positive y-axis in the ...
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vote
2answers
2k views

Foci of Ellipse - given: Width and Height

Can you help me out with the next problem. I have an ellipse based on a width and a height. Is there any way you can find out where the focal points are? I need this information because I need to ...
2
votes
2answers
2k views

Proof that the Convex Hull of a finite set S is equal to all convex combinations of S

In $C^n$, how would you prove that the convex hull of a finite set $S$(convex hull being the intersection of all convex sets which contain $S$) is equal to the set consisting of all convex ...
0
votes
3answers
311 views

The equation of an ellipse

I have a couple of questions regarding ellipses. Get the equation of the ellips With Foci $(\pm 3,0)$ and which goes through $(2,\sqrt{2})$. This one I didn't understand AT ALL. I need some ...
1
vote
1answer
208 views

Equation of a parabola: Translations and directrixes

Find the equation of the paraboles, with: Focus $(3,0)$ and $x=-3$ is the directrix Focus $(0,2)$ and $y=-2$ is the directrix Vertex (I believe it is the vertex, the lowest/highest point) $(1,2)$ ...
2
votes
2answers
116 views

Find out for which values of $\lambda$ the points of the line are inside the circle

We have a line (in parameter): $ x = 2\lambda $ $ y = 1-\lambda$ Find out for which values of $\lambda$ the points of the line are inside the circle of $x^2+4x+y^2-6y+5=0$ What I did: I rewrote ...
0
votes
3answers
58 views

Find the lines which have a certain distance from a certain point

We have a point $P(1,7)$, get the equations of the 2 lines which have a distance of $5$ from point $P$. Both of the lines go through the origin. So I used the formula ...
3
votes
2answers
236 views

Geometry - Area of Siamese Triangles

How can I find the Area of this figure? It is quite curious because it is a particular case of this sequence: Anyone know how to find the area of this sequence as a function of the number of ...