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

learn more… | top users | synonyms (1)

0
votes
1answer
30 views

defining a closed curve in cartesian coordinates

I am trying to implement a track in cartesian coordinates, such that X and Y coordinates are accepted and those are linearly interpolated. The problem is, I want to include circular shapes on ...
0
votes
1answer
31 views

Coordinates of a vertex of a triangle?

Here is the problem: There is a triangle with vertices $A,B,C$ in a cartesian coordinate system, where coordinates of points $A$ and $B$ and the angle $\alpha=\measuredangle ABC$ are given. The ratio ...
0
votes
0answers
20 views

Simplest way to calculate the width of a segment of a convex shape

A convex shape $C$ is cut using a a chord. What is the width of the resulting segment? This is the length of the green thick short line in the figure below: Here is my current solution: Mark the ...
6
votes
2answers
109 views

How big is a tetrahedron?

Let $T$ be a tetrahedron with volume $vol(T)$ and edge lengths $a,b,c,d,e,f$ and let $sum(T) = a^3 + b^3 + ... + f^3$. We wish to compare $vol(T)$ with $sum(T)$. [ IMO (1961 #2 ) handles the case of ...
1
vote
0answers
38 views

Finding the point on an ellipse most distant from a given line

$\mathrm C$onsider an ellipse with the origin as its centre, i.e., of the type $$\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1$$ and a line joining two points on the ellipse. $\mathrm T$he problem is to ...
1
vote
2answers
104 views

Intersection of a cone and a plane.

I need a proof that the intersection of a cone with a plane parallel to the cone's axis is a hyperbola.
0
votes
1answer
63 views

How do you 'rotate' a polynomial?

I have a polynomial equation: $$y=(-5 \times 10^{-6} \times x^3)+(0.0004 \times x^2)+(0.0582 \times x)-0.4397$$ Is it possible to "rotate" this polynomial curve (maintaining the shape) around the ...
0
votes
1answer
24 views

Find the images of (1,0) under reflection in L?

Consider the line $$L = \{(x,y): x - 2y = 2\}$$ Find the images of $(1,0)$ under reflection in $L$? Thanks in advance.
1
vote
2answers
62 views

Indefinite integral with sector of ellipse

An ellipse is given by the following equation: $$ 152 x^2 - 300 x y + 150 y^2 - 42 x + 40 y + 3 = 0 $$ After solving for the midpoint we have: $$ 152 (x-1/2)^2 - 300 (x-1/2) (y-11/30) + 150 ...
0
votes
1answer
32 views

doubt with direction angles

Is it possible for a 3D vector to be drawn with the direction angles of $\alpha=45^\circ$ and $\beta=45^\circ$ ? if yes what is measure of $\gamma^\circ$? I calculated $\cos^2(45^\circ ...
-1
votes
0answers
37 views

Proof in analytic geometry using vector multiplication

Let us have a triangle $\Delta ABC$. $H$ is the intersection of heights if and only if $$ \overrightarrow{HA}\cdot \overrightarrow{HB} = \overrightarrow{HB}\cdot \overrightarrow{HC} = ...
1
vote
1answer
29 views

Algebraic step on a trig expressiom in linear algebra

$$W = ||V||(\cos(\varphi)\cdot \cos(\theta) - \sin(\varphi)\cdot\sin(\theta), \cos(\varphi)\cdot\sin(\theta) + \sin(\varphi)\cdot\cos(\theta))$$ $$= (v_1 \cos(\theta) - v_2 \sin(\theta), v_1 ...
1
vote
0answers
64 views

Find max distance from $(0,0)$ to line defined on ellipse.

I have got a following problem : $E = \{ \frac{x^2}{a^2} + \frac{y^2}{b^2} =1 \}$ $N$ - line (normal) perpendicular to E at $(x_0,y_0)$ Find max $dist(N,(0,0))$ So I am starting with attempt to ...
0
votes
2answers
48 views

Questions about elipse

Given the center of an elipse and three of its points, is this elipse completely determined? What is the easiest way to show that five points of an elipse are enough to determine the elipse?
2
votes
2answers
62 views

Find intersection points of a line with a circle, and the equation of another circle passing through those points [closed]

If the line $x=2y$ meets the circle $x^2+y^2-8x+6y-15=0$ at points $P,Q$, find the co-ordinates of $P$ and $Q$ and the equation of the circle passing through $P,Q$ and at the point $(1,1)$. Could I ...
1
vote
1answer
42 views

How to find a point at a certain distance to other points on the same line

Assuming the points A(x1,y1) and B(x2,y2) and distances between AB (d1) and AC (d2) are known. How can I find the point C(xp,yp)? Actually it has a trivial solution, writing the distance equation 2 ...
0
votes
2answers
48 views

Find a Cartesian Equation for the Plane Satisfying Those Properties

Find the Cartesian equation of the plan parallel to j and passes through the intersection of the planes described by the equations x + 2y + 3z = 4, and 2x + y + z = 2. I was able to get the ...
0
votes
0answers
21 views

What is the analog of the scalar triple product in two dimensions?

Is there a standard name and/or a notation for the analog of the scalar triple product in two dimensions? Namely, i am interested in the following operation: given two elements $\vec u$ and $\vec v$ ...
3
votes
1answer
80 views

Triangle, Circle Problem

What is the area $\triangle DEF$ ? I solved it using analityc geometry. I want to see if there is way to solve it using plane geometry. I did it: $x^2+y^2=400$ $(x+10)^2+y^2=100$ I found the ...
0
votes
0answers
22 views

Plane and symmetrical lines

I need to solve this problem (sorry for bad english) I have plane $\pi$ and line $p_1$ intersecting $\pi$ in point $P$. Then I find line $p_2$ symmetrical to line $p_1$ where $\pi$ is plane of ...
0
votes
1answer
33 views

Locus of intersection between $y= 8\lambda/(\lambda ^2 + 4)$ and $y =2 \lambda x/(4-\lambda^2)$

I have the equations $$y=\frac{4\lambda}{\frac{1}{2}\lambda^2+2}\quad \text{and}\quad y=\frac{\lambda x}{-\frac{1}{2}\lambda ^2 + 2}$$ each representing a line. I'm asked to find the locus of the ...
1
vote
2answers
29 views

How to interpret the equation of a line in 3D through two points, when there are $0$s in the denominator? [closed]

If $A=(0,0,0)$ and $B=(1,0,0)$ are two points of a line in three dimensions, I think its equation should be $$\frac{x-0}{1}=\frac{y-0}{0}=\frac{z-0}{0}\tag1$$ according to the formula ...
0
votes
1answer
26 views

How to show existence of an orthogonal map?

I want to show that the following holds: Let $x,y\in \mathbb{R}^n\setminus\{0\}$ be given and such that $\|x\|=\|y\|$. There is an orthogonal map $T$ such that $Ty=x$ (a rotation). How could one ...
10
votes
5answers
391 views

Circle radius as variable

I am confused. How is $y^2 + x^2 =3x$ a circle? Can someone please help me try to understand why the above a circle, or is it just a typo?
0
votes
3answers
33 views

Find the equation of the plane that contains:

Find an equation for the plane containing the lines $$x = 5y = \frac{z + 1}{4}$$ and $$\begin{cases} x = t \\ y = 2t\\ z = 6t − 1 \end{cases}.$$ I know that finding two points will allow me to find ...
2
votes
1answer
48 views

Finding a line through 4 other lines!

This one's probably easy, but I'm dreadfully stuck and can't seem to figure out a decent method. I have the following lines: $$a: \vec{x}(\lambda)= \left( \begin{array}{ccc} 4 \\ -2 \\ -2 ...
2
votes
2answers
35 views

Showing that normal line passes through a point.

I need to show that a line passes through a point. How should I go about doing this? The question is: Let $L$ be the normal line at $(1,1,1)$ to the level surface of $f(x,y,z) = x^2 - z$ that ...
2
votes
2answers
80 views

What is the Equation for a straight line in a 3D space? And how to find other parelell lines to it?

We know $y=mx+c$ is the equation for a straight line in a 2D graph. And the parallel line that goes through $(x_1,y_1)$ is $y=mx+(-mx_1+y_1)$. But how do we display the straight line in a 3D graph ...
0
votes
4answers
214 views

Show that if an ellipse and a hyperbola have the same foci, then at each point of intersection their tangent lines are perpendicular.

I have to show that: If an ellipse and a hyperbola have the same foci, then at each point of intersection, their tangent lines are perpendicular. So I know that if I prove it for one of the ...
3
votes
1answer
73 views

Is every smooth $\mathbb{R}$-variety isomorphic to an affine variety?

I sadly don't know anything about formal GAGA yet, but I am at least trying to follow my intuition as often as possible. In differential geometry we know that we can embedd every smooth ...
0
votes
2answers
136 views

Smallest circle enclosing three disjoint circles

Consider three disjoint circles not necessarily of same radii. How do you draw the smallest circle enclosing all these three circles? Where is its centre, and what is its radius?
1
vote
0answers
21 views

Finding coordinates of ground-zero with seismic sensors

At the unknown t0 time an explosion occurred at an unknown point X,Y on the 2D plane. We ...
0
votes
1answer
133 views

Equidistant points on a circle

I would like to obtain/generate points on a circle in Cartesian coordinates such that the distance between two consecutive points will be always equal. For example, plotting a circle with radius 100 ...
2
votes
0answers
71 views

how to find angle between two added up vectors in cartesian space

I would like to find the angle between two vectors (theta) -> v1 From i to i+1 v1=(xi1-xi , yi1-y1) and v2 from i+1 to i+2 v2=(xi2-xi1, yi2-yi1), which are shown as in the figure (but v1 and v2 can be ...
2
votes
4answers
502 views

How to find coordinates of reflected point?

How can I find the coordinates of a point reflected over a line that may not necessarily be any of the axis? Example Question: If P is a reflection (image) of point (3, -3) in the line $2y = ...
0
votes
1answer
25 views

help needed in understanding general conics proof

The origin is a centre of a general conic of second degree iff the coefficients of linear terms vanish. $ (\Rightarrow)$ part: Let $$ Q(x,y)\equiv ax^{2}+2h xy+ by^{2} + 2gx+2fy+c=0$$ books ...
4
votes
2answers
102 views

Coordinates of the intersection of two tangents to a circle

Let $A = (x_A, y_A)$ and $B = (x_B, y_B)$. Let $\gamma$ be a circumference of radius $r$, centered in $(0, 0)$; $A$ and $B$ lie outside of $\gamma$, and on the same side of some line $L$ through the ...
3
votes
1answer
61 views

What is the number of intersections of diagonals in a convex equilateral polygon?

Question: [See here for definitions]. Consider an arbitrary convex equilateral polygon with $n$-vertexes ($n\geq 4$) and the $n$-sequence $\langle \alpha_i~|~i<n\rangle$ of its angles which ...
0
votes
3answers
47 views

$x^2+y^2=5$ and point $(-4,3)$. Find the equations of the tangents to the circle and the point.

$x^2+y^2=5^2$ and point $(-4,3)$. Find the equations of the tangents to the circle and the point. This question came up in class and we were unsure of how to do it. Our class spent a good 20 minutes ...
0
votes
1answer
56 views

What is the equation of line that passes through two points ??

Th equation of a straight line that passes through point A(mid point (2,3) and (-8,15) )and point B (that lies 1/3 way from (-1,0) to (4,11) is given by ?? actually I am confused!! calculated the ...
4
votes
1answer
30 views

Minimum dimension to hold $N$ points with given distances?

Suppose you're given $N$ points along with an $N\times N$ matrix $D$ with entries $d_{ij}$ giving the distances between the points (assume that the $d_{ij}$ satisfy the usual requirements of a ...
14
votes
1answer
246 views

Connected unbounded sets $S\subset \Bbb{R}^n$ such that $x\mapsto ||x||^t$ is uniformly continuous on $S$?

Spending the night perusing my old answers, and this question left me wondering about the following. Let's equip $\Bbb{R}^n$ with the usual Euclidean metric, and let us consider the map ...
0
votes
1answer
19 views

3D Vector Equation

Consider the points $A (1, 5, 4)$, $B (3, 1, 2)$ and $D (3, k, 2)$, with $\overline{AD}$ being perpendicular to $\overline{AB}$. (i) Find $AB$ (ii) $AD$ , give the answer terms of $k$. Show that ...
1
vote
1answer
87 views

Finding circle with two points on it and a tangent from one of the points

Two points P1(x1,y1) and P2(x2,y2) are known. In addition, a line slope passing through P1 is known. The aim is to construct a circle (or circular arc) that it passes through both P1 and P2 and it is ...
2
votes
4answers
136 views

Given two points, find another point a perpendicular distance away from the midpoint

I am a computer programmer and need to find the x and y coordinate of a point that is a defined perpendicular distance from a midpoint. For reference, I have tried to attached an image for reference. ...
3
votes
3answers
120 views

Ellipse $3x^2-x+6xy-3y+5y^2=0$: what are the semi-major and semi-minor axes, displacement of centre, and angle of incline?

Given the ellipse $$3x^2-x+6xy-3y+5y^2=0$$ find the following: semi-major axis, $a$ semi-minor axis, $b$ displacement of centre from origin (or coordinates of centre of ellipse $(h,k)$) angle of ...
0
votes
3answers
36 views

Parametric equation with plane equation given

Let $2x + y + z = 2$ be a plane in space. Find the parametric equation of a line of your choice lying in the plane. I find $n=<2,1,1>$ but I need a point to complete.
4
votes
1answer
51 views

Compute direction of a cylinder by using 10 coefficients

I am wondering if anyone knows how to compute the direction of a cylinder by using the 10 coefficients. For example, we have the equation of a cylinder as ...
0
votes
1answer
20 views

Gaussian sums values

I have the following problem: Denoting $S(q,a,\chi ) = \sum_{x=1}^q \chi (x) e(ax/q)$, where $\chi $ is an arbitrary character modulo $q$, I have to prove $$\sum_{a=1}^q \vert S(q,a,\chi ) \vert ^2 = ...
0
votes
1answer
29 views

Deal with non standard form of conic

I want to know how can I calculate latus rectum, tangent at vertex, vertex and axes of a parabola whose equation is not standard. For example, the parabola: $$ 4x^2 - 4xy + y^2 - 10 y - 19 = 0 $$