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

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2
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1answer
28 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?! ...
1
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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
votes
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
22 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
vote
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: $$x = -2 + t$$ $$y = 1 - t$$ $$z = 1 + 2t$$
1
vote
1answer
41 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 ...
1
vote
3answers
72 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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0answers
24 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?
0
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2answers
28 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
votes
2answers
99 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
12 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
1
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0answers
23 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
votes
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 ...
0
votes
2answers
27 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 ...
1
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0answers
29 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
40 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 ...
0
votes
0answers
31 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
45 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
37 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
votes
2answers
55 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
23 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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0answers
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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0answers
37 views

Error Distribution of Canny's algorithm in some borders?

Assume you have two circles which are filled with many ellipses of non-arbitrary size from a finite set. How can you deduce the distribution of the difference of circles' diameters/areas in theory? ...
0
votes
1answer
48 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
90 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
14 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 ...
1
vote
2answers
42 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
34 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
47 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 ...
0
votes
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
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3answers
53 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
27 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 ...
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0answers
22 views

$ABCD$ has area $9$. $M$ is in the middle of $AB$ and the edge $BF$ of length $2$ forms an angle of $60º$, Calculate $[CM,CB,BF]$.

$ABCD$ has area $9$. $M$ is in the middle of $AB$ and the edge $BF$ of length $2$ forms an angle of $60º$. Calculate $[CM,CB,BF]$, knowing that $\mathbb{V}^3$ is oriented by a positive basis. ...
1
vote
1answer
39 views

Prove that $x^2-y^2+xy-1=0$ is a ruled surface

I am studying for an analytic geometry, final but I am totally lost for this problem... We didn't even cover this topic in class (my prof didn't show up for class for two weeks) and I have no clue on ...
1
vote
1answer
78 views

Find the equation of a cylinder

Find the equation of the cylinder that has directrix the curve: $x(t)=t, y(t)=t^2/2, z(t)=0$ and the generatrix is parallel to the line $${x-1\over 1}={y+2\over 1}={z\over 3}$$ I would really ...
0
votes
1answer
52 views

What is wrong with my solution to this problem?

The base $ABCD$ of the figure has area $9$. The point $M$ divides the segment $AB$ on ratio $2$ and the edge $BF$ of length $2$ forms an angle of $60º$. Calculate $[CM,CB,BF]$, knowing that ...
6
votes
4answers
352 views

Coordinates of the center of the circle

I am stuck on this problem: If the lines $y=x+\sqrt{2}$ and $y=x-2\sqrt{2}$ are two tangents of a circle and $(0,\sqrt{2})$ lies on this circle then what is the equation of the circle? I ...
1
vote
0answers
18 views

Analytic structures on $S^1$|

I am currently studying Haefliger's paper "Homotopy and Integrablity". During the last chapter, he applies his theory of $\Gamma$-structures to analytic codimension $1$ foliations. Throughout the ...
0
votes
2answers
52 views

Finding the equation of a circle through 3 points under given conditions.

This question has me stuck at the very beginning and I dont understand what to do. Dont need the solution, just a hint on what to do. Q.A and B are points in the xy plane, which are 2sqrt2 units apart ...
3
votes
2answers
66 views

How to find the equation of diameter of a circle that passes through the origin?

So this was a question that I was solving that got me stuck. Its as follows: Q. Find equation of diameter of the circle $x^2 + y^2 - 6x + 2y = 0$ which passes through the origin. Now I have tried the ...
6
votes
1answer
53 views

What does a linear equation with more than 2 variables represent?

A linear equation with 2 variables, say $Ax+By+C = 0$, represents a line on a plane but what does a linear equation with 3 variables $Ax+By+Dz+c=0$ represent? A line in space, or something else? On ...
4
votes
2answers
47 views

The four straight lines given by the equation $12x^2+7xy-12y^2 =0$ and $12x^2+7xy-12y^2-x+7y-1=0$ lie along the side of the?

I know these equations are called general equation of second degree and also represent a pair of straight lines. I could extract lines from the equation $$12x^2+7xy-12y^2 =0 $$ (these are $$ 3x+4y=0$$ ...
0
votes
1answer
39 views

vertices of a hyperbola the silliest question ever

I'm given that the center of the hyperbola is $(2,1)$ and $a=3$ and asked to find the vertices. Since vertices are on the same line with the axis of symmetry I thought the coordinates should be $(2,1 ...
4
votes
2answers
62 views

The lines $x+2y+3=0$ , $x+2y-7=0$ and $2x-y+4=0$ are sides of a square. Equation of the remaining side is?

I found out the area between parallel lines as $ \frac{10}{\sqrt{5}} $ and then I used $ \frac{|\lambda - 4|}{\sqrt{5}} = \frac{10}{\sqrt{5}} $ to get the values as $-6$ and $14$ . I am getting the ...
0
votes
0answers
19 views

What's the relation between 2 points from 2 different planes?

I'm trying to find the relation between my "text" objects, and my "world" objects. This may be related to development, but I thought this question was better fit for this exchange. I have two ...