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

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39 views

A Statement About Points in the Real Euclidean Space

Suppose that $n \geq 3$, $x$, $y \in \mathbf{R}^n$, $d \colon= |x-y| > 0$, and $r>0$. Then how to prove the following assertions: (a) If $2r>d$, there are infinitely many $z \in ...
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3answers
723 views

Find the length of this chord.

I've been trying to solve this geometry question for past 2 hours but haven't got the answer yet. There are two concentric circles or radius $8 cm $ and $13 cm$ with the common center $O$. $PQ$ is ...
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0answers
183 views

How to generalise a result regarding intersections of cones and other convex sets?

To test for a particular property of positive LTI systems using feasibility problems I've come across the following claim which, intuitively, I believe can be generalised. I think I've (rather ...
-1
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1answer
261 views

Show that the equilateral triangle has congruent angles?

This Question is of Chapter "Straight Line" the diagram of this question shows the values of ABC I am confused abut the values of C(x,y) it should be (b,c) but it's written something else can someone ...
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1answer
367 views

In an equation that looks like the standard form of an ellipse, what must the constant on the RHS equal for exactly one solution?

I am working on a homework question: What must be the value(s) of $c$ for the following equation to have exactly 1 solution? The equation is of the standard form of the equation for an ellipse, ...
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4answers
160 views

closest point to on $y=1/x$ to a given point

I feel like I'm missing something basic - given a point $(a,b)$ how do I find the closest point to it on the curve $y=1/x$? I tried the direct approach of pluggin in $y=1/x$ into the distance formula ...
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2answers
115 views

Find minimum distance

I came across this problem in a maths exam. I solved this by taking that a light ray passes in such a way that it takes least path. But as this was a maths exam, i was wondering if it can be solved ...
2
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3answers
563 views

Simultaneous Equations and Vectors

The question I am currently working on is, "...find $a$ and $b$ such that $\vec{v} = a \vec{u} + b \vec{w}$, where $ \vec{u} = \langle 1,2 \rangle$, $\vec{w} = \langle 1,-1 \rangle$, and $\vec{v} = ...
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2answers
64 views

Finding the equation of a plane.

How do I find the equation of a plane given by the points (0,1,1), (1,0,1) and (1,1,0)? Graphing it, it's a triangle when you connect the points. Can I use this somehow?
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4answers
2k views

What is a point?

In geometry, what is a point? I have seen Euclid's definition and definitions in some text books. Nowhere have I found a complete notion. And then I made a definition out from everything that I know ...
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1answer
148 views

How can I find the intersection of a line vector and a plane?

Here is my vector: $(-3,1,-4)+r(4,0,1)$ And my plane: Created from the following vectors: $x: (3,0,1)+t(-1,1,2)$ $x: (0,2,-1)+s(2,-2,-4)$ $(3,0,1)+t(-1,1,2)+n(2,-2,-4)$ (Cartesian: ...
8
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1answer
202 views

Maps of $\mathbb{R}^3$ preserving the cross product

Given a map $\phi:\Bbb R^3 \rightarrow \Bbb R^3$ such that for all $a,b \in \Bbb R^3$: $$\phi(a \times b)=\phi(a) \times \phi(b)$$ Is $\phi$ necessarily a rotation around the origin or the map ...
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2answers
131 views

How to find angle of plane $7x+13y+4z = 9$ with $xy$ coordinate plane?

How can I calculate inclination of $7x+13y+4z = 9$ with $X-Y$ plane As for as I understand from question is that the angle of plane $7x+13y+4z=9$ with $ax+by+0z=d$ for $(XY)$ plane.
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2answers
486 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 ...
2
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1answer
234 views

Tangent cone to a subset of $\mathbb{R}^3$

Well, I have the set $X=\{(x,y,z) \in \mathbb{R}^3 | 3x^2+2x^3+y^2+z^2=1\}$ How can I calculate the tangent cone at the point $(-1,0,0)$ ? What are the standard ways to calculate the tangent cone to ...
4
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0answers
92 views

Points at Integer Distances in 3-space

With the restriction no three points in a line, no four points on a circle, there is a 7 point configuration of points on the plane such that all pairs of points are at integer distances. [1] For ...
2
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2answers
187 views

Given an algebraic curve $F(x,y)=0$, why do the partial derivatives of $F(x,y)$ being zero at a point imply the plane curve has a singularity?

I'm looking at algebraic plane curves of the form $F(x,y)=0$ and trying to figure out why for points on the curve such that $\frac{\partial F}{\partial x} = \frac{\partial F}{\partial y}=0$, the plane ...
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2answers
32 views

Computation with scalar product

Let $\vec{a}$ and $\vec{b}$ be vectors from $V_3$. Suppose, that $|\vec{a}| = 1$, $|\vec{b}|=2$ and the angle between $\vec{a},\vec{b}$ is $\frac{\pi}{3}$. Use the properties of scalar product and ...
0
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1answer
104 views

Normal vector to surface

This is a very noob question, but can someone please give me an example of finding the normal vector to a surface (if this is the word in English) which is defined by three points in it. I know that ...
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3answers
3k views

how to find focal radius in parabola?

will we find focal radius in parabol, if our equation is $y^2=12x$. Do I need another variable? I have tried many times but I cannot find this problem. Thanks.
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2answers
85 views

Area of a decentered circunference [duplicate]

Possible Duplicate: Area of a portion of an arbitrarily-placed circle? Given a circunference of radius $R$ with the center in $P\equiv(x_0,y_0)$ $$(x-x_0)^2+(y-y_0)^2=R^2$$ I need to know ...
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1answer
257 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 ...
2
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1answer
192 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} ...
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1answer
143 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
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1answer
62 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 ...
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2answers
585 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
116 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
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2answers
3k 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 ...
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4answers
10k 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
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1answer
374 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 ...
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1answer
438 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.
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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
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2answers
533 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 ...
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1answer
700 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
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4answers
140 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
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1answer
194 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} ...
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3answers
887 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 ...
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2answers
187 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
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3answers
123 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 ...
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1answer
515 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
336 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 ...
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1answer
157 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 ...
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1answer
100 views

Lines which intersect the postive half axis of x

We have to find out which lines intersect the postive half axis of x. According to this formula we can determine if the angle between two points(A[x1,y1] and B[x2,y2]) of the line (angle A0B where 0 ...
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0answers
418 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 ...
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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 ...
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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 ...
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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
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1answer
266 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} ...
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3answers
229 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
595 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 ...