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

learn more… | top users | synonyms (1)

0
votes
3answers
36 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
54 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 \end{...
2
votes
2answers
66 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
2k 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
743 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
102 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 $\mathbb{R}$-...
0
votes
2answers
382 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
48 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
436 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
126 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
5answers
2k 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 = x+...
0
votes
1answer
33 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
367 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 ...
2
votes
1answer
1k 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
56 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
66 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
36 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
288 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 $N_t:\Bbb{R}^...
0
votes
1answer
26 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
151 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 ...
3
votes
4answers
913 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. ...
4
votes
3answers
220 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
61 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
63 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 $$c_0+c_1x+c_2y+c_3z+c_4x^2+c_5y^2+c_6z^2+...
0
votes
1answer
27 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 = ...
1
vote
1answer
50 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: $$ 4x^2 - 4xy + y^2 - 10 y - 19 = 0 $$
3
votes
0answers
143 views

Calculate the flux of $\underline{v}$ across the boundary of the sector.

For $a\in(0,1)$, calculate without use of the divergence theorem the flux of $\underline{v}(x,y) = g(y/x)(-1/x,1/y)$ across the boundary of the sector $ S_a := \{(x,y)\in \Omega : 1\leqslant x^2+y^2 ...
0
votes
1answer
207 views

Find the equation of the parabola with its vertex on the line $2y-3x=0$?

Its axis of symmetry is parallel to the x-axis, and it passes through the two points $(3,5)$ and $(6,-1)$
0
votes
1answer
37 views

Paramaterizing a Parabola with $3$ points.

Let $A, B, C$ be vectors in $\mathbb R^2$. I want to show that the set $\{A+tB+t^2C\mid t\in\mathbb R\}$ defines a parabola in $\mathbb R^2$, but I'm having a hard time doing so, since I can't solve ...
5
votes
1answer
96 views

What is the equation of the reflections of a fixed point across all the tangents to a fixed circle?

Given a fixed circle "c" and a fixed point "A" (in the plane of the circle), draw the tangent to the circle at a variable point "X" (movable, but constrained to be on the circle), reflect "A" across ...
0
votes
0answers
68 views

What is the equation family of the projectile-motion-with-air-resistance eqn?

The general form of the equations of projectile motion with air resistance are (from here) $s_y(t) = -\frac{mg}{k}t + \frac{m}{k}\left(v_{yo} + \frac{mg}{k}\right)\left(1 - e^{-\frac{k}{m}t}\right)$ ...
0
votes
0answers
617 views

Asymmetric hyperbola-type curve? (for fitting to data)

I have this question: what would be the name and equation of a curve which resembles a parabola but has not the requirement of symmetry? So the general parabola equation is: $y=ax^2+bx+c$ I must ...
17
votes
2answers
1k views

How to generate points uniformly distributed on the surface of an ellipsoid?

I am trying to find a way to generate random points uniformly distributed on the surface of an ellipsoid. If it was a sphere there is a neat way of doing it: Generate three $N(0,1)$ variables $\{x_1,...
0
votes
2answers
73 views

Equation of circle in terms of length of arc above $x$-axis

Say I have a circle centered at $(0,b)$ that passes through $(-5,0)$ and $(5,0)$ and has upper-half length $d.$ Now I've figured out that the equation of the circle is $$x^2 + (y-b)^2 = 5^2 + b^2$$ ...
0
votes
1answer
84 views

Parabola max. number

If the directrix and the tangent at vertex of a parabola are given then what is the maximum number of parabolas that can be drawn? Well according to me the answer should be 1 because the distance ...
4
votes
4answers
268 views

Equation of a line tangent to circumference

Discover the general equation of the tangent line to the circumference $x^2 + y^2 - 2x + 4y + 1 = 0$ by the point $(3,4)$. NO CALCULUS. by the circumference equation i discovered that $C(1, -2)$...
1
vote
2answers
460 views

implicit equation for elliptical torus

I just wondering what the implicit equation would be if an ellipse with major axis a and minor axis b, rotating about the Z axis with a distance of $R_0$. The $R_0$>a and $R_0$>b which means the ...
0
votes
1answer
120 views

How to find the equation of conics given foci and directrices

Find the equations of the following conics, each with its centre at the origin. (a) A hyperbola with foci $(\pm4, 0)$ and directrices $x= \pm2$ (b) An ellipse with foci $(0, \pm4)$ and directrices $...
2
votes
3answers
90 views

Equation of line passing through point.

The straight line $3x + 4y + 5 = 0 $ and $4x - 3y - 10 = 0$ intersect at point $A$. Point $B$ on line $3x + 4y + 5 = 0 $ and point C on line $4x - 3y - 10 = 0$ are such that $d(A,B)=d(A,C)$. Find ...
1
vote
0answers
18 views

geometry of a hyperbola and circle drawn together

how to calculate the radius of a circle which is drawn below(inwards) the hyperbola curve touching it.need a relationship between these hyperbola and circle .If a circular object is place below the ...
0
votes
3answers
88 views

Find a vector minimizing the distance from set

Find a vector $\Pi_Z(x)$ minimizing the distance between $x=(5,10)\in\mathbb{R}^2$ and set $Z=\{(x,y)\in\mathbb{R}^2:x\ge0, y\le\sqrt{x}\}$
0
votes
2answers
42 views

Is there a function whose graph is contained in one quadrant of the coordinate plane?

Is there a function whose graph is contained in one quadrant of the coordinate plane? It should be related to maths and not physics. Please give me the equation and if possible its picture.
2
votes
1answer
84 views

How to find the equation for the line $t$, in the plane $\pi$ and concurrent to other 2 lines

The exercise says that $t$ is in the plane $\pi: x-y+z =0$ and is concurrent to the lines: $$r:\\x+y+2z=2\\x=y$$ and $$s:\\z=x+2\\y=0$$ I've transformed $r$ to the form: $$r:\\x = \lambda\\ y = \...
0
votes
1answer
16 views

Find the line $t$ that is concurrent to $r$ and $s$ and parallel to $MN$

I need to find the vector equation for the line $t$ that is concurrent to both: $$r:X = (1,1,-1)+\lambda(2,1,-1)$$ and $$s:\\x+y-3z = 1\\2x-y-2z=0$$ And also, $t$ is parallel to $MN$ when: $$M = (1,-...
0
votes
1answer
80 views

Find the vector equation of the line parallel to the plane $\pi$, perpendicular to the line $AB$ and that intercepts $s$

I have the plane: $$\pi:2x-y+3z-1 = 0$$ $$A = (1,0,1), B = (0,1,2)$$ And $$s: X = (4,5,0) + \lambda (3,6,1)$$ I need to find a line that is perpendicular to $AB$, parallel to the plane $\pi$ and ...
0
votes
1answer
82 views

Find the equation of the plane that contains the line $r$ and makes an angle with $s$

I have the line: $$r:\\3z-x = 1\\y-1 = 1$$ And the plane makes an angle $\theta = \arccos \frac{2\sqrt{30}}{11}$ with the line: $$s:X = (1,1,0) + \lambda(3,1,1)$$ What I tried: From the equations ...
0
votes
1answer
34 views

Vectors - theory on cross product

If $X$ is a point on a line through $P$ and $Q$, $X=OX, P=OP, Q=OQ$ (all are vectors but $X$) then: $$X \times (Q-P) = P \times Q$$ I subbed in the $OX$, etc and simplified, but did not get each ...
1
vote
1answer
70 views

Why this isn't working? Find the points of the line $r$ that has the distance $\sqrt{\frac{14}{3}}$ from line $s$.

I have the line $$r:\\x+y=2\\x=y+z$$ and $$s:x=y=z+1$$ I need to find the points of $r$ that has distance $\sqrt{\frac{14}{3}}$ from $s$. What I tried: By using the formula for distance of a ...
1
vote
2answers
532 views

How to find the number of squares formed by given lattice points?

Let us say that we are N integer coordinates (x, y) - what would our approach be if we were supposed to find the number of squares we could make from those given n points? Additionally, if we were to ...
1
vote
1answer
51 views

Finding circumcenter

In a triangle A(1,2) B(2,3) C(3,1) and $\angle A = cos^{-1}(4/5)$, $\angle B = \angle C = cos^{-1}(\frac{1}{\sqrt{10} }) $ Ordinate of circumcentre of the $\triangle$ is ? I have tried solving by ...