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

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2
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1answer
58 views

What geometric object is given by this equation?

What geometric object is given by this equation? $x^2+y^2+z^2+2xy+2xz+2yz-x-y-z-6=0$ Maple says it's a hyperboloid of one sheet, but is there a way to show it without going the long way by using the ...
3
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2answers
90 views

Let $ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$. Determine $S+S+…+S $.

Let $$ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$$ By the usual notation for sum of sets let $$ 2S\overset{\text{not}}{=}S+S=\{(x_1+x_2,y_1+y_2) \ | \ (x_1,y_1), ...
6
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0answers
53 views

Is this solution legal?

Let $M(1,-1)$ be a point in a plane. Find its distance from a line given by $x+2y-4=0$. Later on I found a formula: $$d=\frac{\left | Ax_{0}+Bx_{0}+C \right | }{\sqrt{A^2+B^2}}$$ But I did it ...
0
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0answers
49 views

Algorithm to calculate line segments between two points bounded by multiple surfaces

Problem statement: As a specific case, let's say I have a volume composed of a series of concentric cylinders. Given a fixed point P (a,b,c), and another randomly sampled point Q (x0,y0,z0), I would ...
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0answers
33 views

Proof regarding hyperbolas

Given the parameters $a,b>0$ we set $c:=\sqrt{a^2+b^2}$ and $e:=\large\frac{c}{a}$ (eccentricity), the focal points are $F=(c,0)$ and $F'=(-c,0)$, the directrix $L$ with the equation ...
1
vote
3answers
36 views

Finding extrema.

Find the minimum distance between point $M(0,-2)$ and points $(x,y)$ such that: $y=\frac{16}{\sqrt{3}\,x^{3}}-2$ for $x>0$ . I used the formula for distance between two points in a plane to get: ...
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0answers
12 views

distance between planes in a simplex

In an Euclidean space there are n points at equal distances d to each other (regular simplex). Find out a distance between two parallel planes, one spanned at points numbered 1 through k, the other at ...
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0answers
12 views

4-dimensional simplex

In a 4-dimensional Euclidean space, there is a simplex, with given lengths of all the edges aij = distance(Ai,Aj). Find a distance between gravity centers of sides, opposite to each other. Notice: ...
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0answers
18 views

Why $\|X-F\|=e|(X-F)\cdot N -d|$ should be written as $\|X-F\|=e|(X-F)\cdot N +d|$?

I'm reading Apostol's Calculus. $\quad $ And I've tried to do the following exercise: $\quad \quad \quad \quad $ I am a little confused: I have the portuguese version of the book, and it ...
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0answers
37 views

Help with Apostol's “Calculus, vol. 1”, Section 1.18

In section 1.18 ("The area of an ordinate set expressed as an integral"), Apostol proves two theorems. the first, theorem 1.10, deals with the area of a function's ordinate set; the second, theorem ...
6
votes
1answer
157 views

Definition of a Cartesian coordinate system

Apologies if this is a basic question, but I'd really like to clarify the exact meaning of what a Cartesian coordinate system is. Heuristically, is it correct to say that a Cartesian coordinate system ...
1
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1answer
15 views

Graphical details of changing functions

I'm struggeling a bit with the transformations of a function when values are changed (for instance an offset to the right etc). So far I have found the following: ...
2
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0answers
40 views

Find the equations of the lines tangent to the circle $x^2+y^2=r^2$ that pass through the point $(a,0)$?

Find the equations of the lines tangent to the circle $x^2+y^2=r^2$ that pass through the point $(a,0)$. My book explains that the equation of this line is $y=m(x-a)$ and then we make the ...
0
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2answers
53 views

Questions on the relation of the axis of a cone to its conic sections

(1) Does the axis of a cone pass through the foci of any its conic sections? Consider the image below: Is the intersection of the axis of cone and the ellipse the same as the focus of the ellipse? ...
2
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1answer
29 views

Analytic geometry line segments

This is a very interesting analytic geometry math problem that I came across in an old textbook of mine. It is quite nice and I decided I would share it with MSE for future reference and a fun time?! ...
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2answers
17 views

Why the distance from the point to the line is $\frac{|(P-Q)\cdot N|}{\|N\|}$?

$P$ is a point in the line $L$, $N$ is a vector normal to $L$ and $Q$ is a point out of the line. I know that taking the subtraction of $(P-Q)$, I create a vector that goes from $P$ to $Q$ but I don't ...
2
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0answers
51 views

Is the following a conic section

All vectors are in $\mathbb{R}^3$ and only $\mathbf{r} = \left[ x; y; z \right]$ is unknown. My question is does the following system define a conic section in the $x-y$ plane and, if so, how can I ...
0
votes
1answer
23 views

Location of an arbitrary point of an ellipse

Given this ellipse equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $(a>b>0)$ and $c:=\sqrt{a^2-b^2}$ aswell as the focal points $F=(c,0)$ and $F'=(-c,0)$, why can we say without loss of ...
1
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1answer
34 views

Parametric equations of perpendicular lines

I'm having problems with this: Find the parametric equation of the line that passes through the point $(-1, 4, 5)$ and is perpendicular to the line: $$x = -2 + t$$ $$y = 1 - t$$ $$z = 1 + 2t$$
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1answer
45 views

How many sets of four points in an MxN grid have one point contained by three other points?

Given a 3x3 grid: 1 2 3 8 9 4 7 6 5 We find 126 distinct sets of 4 points $$\binom{9}{4}$$ There are 8 cases such that when the points are connected with a line in clockwise direction, one point ...
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3answers
85 views

Finding the locus of midpoint of $AB$

The normal to the ellipse $b^2x^2+a^2y^2=a^2b^2$ is passing through the x-axis in point $A$ and through the Y-axis in point $B$. Point $P$ is the midpoint of $AB$. Need to find the locus of $P$. ...
2
votes
1answer
36 views

Connected component identification?

Suppose I give a random 2 variable polynomial relation such as: $$x^3+y^3=10$$ $$x^2 + 7yx^4 + x^2-15=0$$ Etc... How do I determine how many individual pieces there are to the graph?
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27 views

Solving euclidean geometry problems with analytical geometry

Can anyone recommend a good resource about applications of analytical geometry in doing elementary geometry problems like ones on IMO?
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2answers
29 views

Identity simplification

How do you express $\dfrac{\sin A\sec A\cot A}{\tan A}$ in terms of sine and cosine? I have simplified using $\sec(A)$ as $\cos^{-1}(A)$ and also $\cot(A)$ as $\dfrac{\cos(A)}{\sin(A)}$, and appear ...
5
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2answers
105 views

What's interesting in latus rectum?

I'm a maths teacher in Italian secondary school and I've been spending some time trying to construct "meaningful" problems about conic sections. I particularly like problems which focus on practical ...
0
votes
1answer
13 views

equation of a line parallel to a given ine at a constant disance?

what is the equation of a line parallel to a given line say y=x at a constant disance of 1 unit from it? I guess there will be 2 equations,one above x axis and other below x axis
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0answers
25 views

Faster Alternative than Calculating Euclidian Distance to determine which Coordinate has Max Distance from a fixed coordinate (eg (0,0))

I am developing a program that needs me to determine which coordinate in a 2-d figure has maximum distance from a fixed coordinate. Let me demonstrate: 3 points: (1,3), (2,2), (5,0) ; Fixed point: ...
0
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1answer
28 views

Given curve is $y=x^2-1$, and $A(0,y_{1}),B(1,y_{2})$. Determine point $M$ between $A$ and $B$ so the area $AMB$ has maximum value.

I have found the equation for line between $A$ and $B$: $$y=x-1$$ Equation for tangent is: $$y=x-\frac{5}{4}$$ Coordinates of point $M(\frac{1}{2},\frac{-3}{4})$ Because the area $AMB$ is ...
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2answers
31 views

How to check a point is inside an ellipsoid with orientation?

For an ellipsoid of the form $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $$ with orientation vector $\vec r$ and centre at point $\vec p$, how to find whether a point $\vec q$ is ...
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0answers
33 views

Intercepted at the Coordinate Axes

A line passes through point $(2,2)$. Find the equation of the line if the length of the line segment intercepted by the coordinate axes of the square root of $5$. The correct answer among the choices ...
2
votes
2answers
41 views

Find the line segments cut off by the plane $ax+by+cz+d=0$ on the coordinate axes, if $abcd\neq 0$

I'm reading Pogorelov's Geometry. Find the line segments cut off by the plane $ax+by+cz+d=0$ on the coordinate axes, if $abcd\neq 0$. Writing the equation as $a(x-x_0)+b(y-y_0)+c(z-z_0)=0$, I ...
2
votes
1answer
48 views

What is the need to define so many forms of equation of a straight line?

When I study maths, I try to understand why the mathematicians brought out this concept or what usefulness they might have seen in the concept that they worked upon. So when I started with straight ...
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0answers
38 views

Why is the ratio of external division of a line by a point negative?

Say there is a line AB externally divided by point C. AC:BC=3:2; then if we are representating it mathematically, we would write it as -3:2 (that's what I think). Now what I am trying to understand is ...
3
votes
1answer
49 views

Surface Area of unit n-sphere covered by rotating a unit vector around a fixed unit vector such that angle between the two vectors is always fixed.

Consider an n-dimensional unit sphere and unit vector from the origin with its tip lying on the surface of sphere. Consider another vector which makes some angle say $\epsilon$ with unit vector. From ...
2
votes
1answer
39 views

Placing $n$ points so that their distances lie in $[1,a]$

What is the maximum number of points we can place in the plane so that the distance between any two such points is in the interval $[1,a]$? I had initially conjectured that the maximum could be ...
0
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2answers
94 views

Find the equation of circle touching given lines and a given point. [closed]

$U: 3x+4y-20=0$ and $v:3x+4y+10=0$ are two straight lines. Find the equation of circles touching the given lines and passing through point $P(1,2)$.
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0answers
19 views

Converting X, Y and Z Co-ordinates(Cartesian Co-ordinate Systems) into their respective angles(Yaw, Pitch and Roll))

I've been having this trouble to convert the vector components X, Y and Z into their corresponding angles. So far I was able to analyse these things. let $acc_x, acc_y$ and $acc_z$ be 3 co-ordinate ...
0
votes
1answer
24 views

Equation of the affine transformation that fixates a certain line

I have to find the equation of the affine transformation of the affine plane $A_2$ that (1) fixates the line $s: x + y - 1 = 0$ and (2) such that $A(Q)=P$, where $Q(1,2)$ and $P(2,1)$. How should I ...
0
votes
1answer
20 views

Question about determinig types of surfaces?

$$x^2 +y^2 +z^2 +2x +1=0$$ This is an equation for dot if we are talking about surfaces, right? It is not an ellipsoid.
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13 views

Finding the second dirextris.

how can I find the equation of the diretrix of the curve of the second order, given both focal points and the other diretrix?
0
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1answer
50 views

How many ellipsoids can be maximally inside a circle?

This discussion is related to this discussion here where I want to deduce the area difference between such two circles filled with ellipsoids. Actually, to understand this difference is the main ...
6
votes
2answers
94 views

How to determine whether a point is inside a closed region or not?

Take the following parametric equation of an implicit curve as an example: $$ \left\{\quad \begin{array}{rl} x=& 9 \sin 2 t+5 \sin 3 t \\ y=& 9 \cos 2 t-5 \cos 3 t \\ \end{array} \right. $$ ...
2
votes
0answers
17 views

Finding the transformation matrix of a projective transformation in RP^2

So I want to understand how to find the matrix that represents the projective transformation that sends 4 given points to 4 given images, I know that 4 points are necessary to determine it but I can't ...
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vote
2answers
50 views

Bisector of two lines in the euclidean space $\mathbb{E}_3$

Let $$r: \begin{cases} x + z = 0 \\ y + z + 1 = 0\end{cases}$$ and $$s: \begin{cases} x - y - 1 = 0 \\ 2x - z -1 = 0\end{cases}$$ be two lines in the euclidean space $\mathbb{E}_3$. It is easily ...
1
vote
1answer
37 views

Verifying if these basis are positive or negative?

Verify if the basis $E=(e_1,e_2,e_3)$ and $F=(f_1,f_2,f_3)$ are positive or negative with: $$f_1=e_1\quad \quad\quad\quad\quad f_2=e_2+e_3\quad \quad \quad\quad \quad f_3=e_1+e_2 $$ I did ...
1
vote
1answer
48 views

Finding the smallest square inside a parabola. [duplicate]

I just thought of a problem earlier today, but wanted to know if there was an easier way of acquiring the answer. Say I have a standard parabola $y=x^2$ with 3 points on it $P,Q,R$ and another point ...
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0answers
21 views

finding the axis of a hyperbolic cylinder

I have data (a lot of points x,y,t) which are modeled by a hyperbolic cylinder $t^2 = b_0+b_1x+b_2y+b_3x^2+b_4xy+b_5y^2$ I know that if i just make a set of 6 equations from it, and than randomly ...
0
votes
1answer
35 views

Geometric proof that (symmetry w/r to $x$ and $y$ axes) $\implies$ (symmetry w/r to origin)

I'm trying to prove that reflecting a point about the x and y axes is equivalent to reflecting it about the origin. Is my proof valid? How could I improve it? Proof: Take a point $a$ in the first ...
1
vote
3answers
56 views

Breaking down the equation of a plane

Could someone explain the individual parts of a plane equation? For example: $3x + y + z = 7$ When I see this I can't imagine what it's supposed to look like.
0
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0answers
30 views

Fit an ellipse with known semi-major-axis and points

In my particular case I am given a projection of a circle onto the $xy$-plane and the radius $r$ of said circle. This results in an ellipse with semi-major axis $a$ equal to $r$. Like in this other ...