# Tagged Questions

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

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### Max. distance of Normal to ellipse from origin

How Can I calculate Maximum Distance of Center of the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$ from the Normal. My Try :: Let $P(a\cos \theta,b\sin \theta)$ be any point on the ...
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### How to calculate the area closed by a parabola and a line without calculus?

In order to simplify the problem, suppose we have a parabola $y=ax^2+bx+c$, here $a\neq0$, and a line $y=kx+d$, here $k\neq0$. We can assume that they will intersect at two different points. Thus, the ...
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### Unit vectors orthogonal to L

I have a line $L$ in $\mathbb{R}^2$ that passes through two points: $u = [9;7]$ $v = [1;-5]$ How do I find all unit vectors orthogonal to $L$? I know: $[x;y] * [8;12] = 0$ and $x^2 + y^2 = 1$ ...
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### Find all unit vectors orthogonal to line with two given points

I have a line $L$ in $\mathbb{R}^2$ that passes through two points: $[9;7]$ and $[1;-5]$ How do I find all unit vectors orthogonal to $L$?
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### With what analytic functions can one construct the $(x,y)$ coordinate axes using a straightedge and a compass?

Given the graph of $y = \frac{1}{x}$, construct the $(x,y)$ coordinate axes using a straight edge and a compass. The solution to this problem is known (mouse over the spoiler text below for a ...
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### 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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### 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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### 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: ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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 ...