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

learn more… | top users | synonyms (1)

3
votes
1answer
21 views

How homogenization of line and curve works?

I am given a curve $$C_1:2x^2 +3y^2 =5$$ and a line $$L_1: 3x-4y=5$$ and I needed to find curve joining the origin and the points of intersection of $C_1$ and $L_1$ so I was told to "homogenize" ...
2
votes
1answer
32 views

Rigorous books on geometry

I am looking for a rigorous book on both 2d and 3d euclidean geometry, and also how analytic geometry can be developed from synthetic geometry. I haven't really found such a book yet. I would be very ...
0
votes
1answer
14 views

How do I determine k so that the line of the beam is parallel to a $60^\circ$ angle?

I have the equation of a beam that looks like this: $$(x + y - 5) + k(2x - 3y) = 0$$ I know that the angular coefficient of a $60^\circ$ angle is equivalent to the root of 3. $$m = \sqrt3$$ Though, ...
0
votes
1answer
49 views

formula for a sphere?

is there such a thing as a formula for a sphere? Is it $x^2+y^2+z^2=1$? if so, does the $1$ denotes a radius of $1$ for said sphere? what are the possible alterations for such a formula?
0
votes
0answers
10 views

Proving That Two Paths of Different Lengths Are Adjoined

In the section on 'Adjoining Paths of its 'Topology' book's page on 'Path Connectedness', WikiBooks shows that, for any topological space $X$ with members $a$, $b$, and $c$, the following…: ...
0
votes
1answer
10 views

Derivation of the Parametric Form of the Equation of a Line From Its Two-Point Form?

Wikipedia's documentation on the parametric form of a linear equation states in the paragraph between two different sets of equations available for use in determining a line's parametric equations ...
0
votes
3answers
41 views

What is the equation of the line that is parallel to the y-axis?

I have a line, parallel to the $y$-axis, that passes through a point, P: $$P(1/2,-3/5)$$ What is the equation of the line? What I tried: $$(y−y_0)=m(x−x_0)$$ $$(y+3/5)=m(x−1/2)$$ ...
-1
votes
0answers
26 views

Find intersection of curve and its asympotetes? [on hold]

The asymptotes of given curve are coming $y = x +1, y = -x +1, y = -1 $ Now how to find their point of intersection?
0
votes
1answer
33 views

How small can an external angle of a circumference be if made of tangents?

Lets imagine the angle ABC where the lines AB and CB are tangents to a circumference which center is C. Lets assume that the points where the line AB touches the circumference is P and the point where ...
2
votes
2answers
58 views

Show that $PF.PG=b^2$ in a hyperbola

If the normal at P to the hyperbola $\frac {x^2}{a^2}-\frac {y^2}{b^2}=1$ meets the transverse axis in G and the conjugate axis in G' and CF be the perpendicular to the normal from the center C then ...
0
votes
1answer
31 views

What are the coordinates of your position?

Suppose you start at the origin, move along the x-axis 3 units. Then face downwards and move forward 4 units. Then turn right and move 7 units. Then (relative to your current position) face downwards ...
1
vote
0answers
36 views

Interesting properties of ellipses

While solving a book on ellipses, I came across the following property of an ellipse which was given without proof:- $\mathbf Property:$If the normals be drawn at the extremities of a focal chord of ...
0
votes
1answer
31 views

Analytic Geometry, tangent plane

Please don't ban this question, I just need some advice on how to find the equation of the tangent plane on a point of the hyperbolic paraboloid which is parallel to a certain plane say ax+by+cz+d=0. ...
0
votes
2answers
40 views

Finding the value of $p$ in the parabola $y^2=2px$

I just started to learn the parabola shape and I have a question: Given the parabola $y^2=2px$ $(p>0)$. The chord $AB$ of the parabola passes through the focus $F(\frac{p}{2},0)$. The slope $m$ ...
0
votes
1answer
18 views

Finding the distance of the line to apoint

Find the distance from $3x-4y-10=0$ to the point $(2,0)$ my answer here is $ \dfrac{-4}{2}$ or $-2$ by substituting the given by the use of the formula but Im just wondering if there's a negative ...
-1
votes
1answer
51 views

Find equations of two circles drawn through the origin which cut another circle orthogonally and touch a line

Find equations of two circles which are drawn through the origin to cut the circle $$x^2+y^2-x+3y-1=0$$ orthogonally and to touch the line $$x+2y+1=0$$. $$x^2+y^2-2ax-2by=0$$----(1) is the general ...
0
votes
1answer
35 views

Meaning of $\dashv\vdash$

I was looking at ProofWiki's articles 'Definition:Equidistance' and 'Definition:Between (Geometry)'and came across the symbol '$\dashv\vdash$.' What does it mean?
1
vote
1answer
34 views

Do we have $proj_u(a) + proj_u(b) = proj_u(a+b)$?

Let $a, b, u$ be vectors in $\mathbb{R}^3$. For two vectors $r, u$ in $\mathbb{R}^3$, let $proj_u(r)$ be the projection of $r$ on the line of $u$ in $\mathbb{R}^3$. Do we have $proj_u(a) + proj_u(b) = ...
3
votes
1answer
48 views

Intersection of a line through two points on a unit circle with real axis

Suppose we are given two points on unit circle which are represented as complex numbers $u$, $v$. We want to show that the intersection of the line through $u$ and $v$ and the real axis is ...
-2
votes
0answers
67 views

The x-intercept of a straight line through points on the unit circle [closed]

Suppose $u$ and $v$ are complex points on the unit circle such that the line through $u$ and $v$ intersect the real axis. If $z$ is the point where this line intersect the real axis, then how can we ...
0
votes
0answers
14 views

Good book for Solid Analytical Geometry?

So my teacher uses this book, William H McCrea's Analytical Geometry of Three Dimensions, but it's awfully hard and dry. I need something with more exercises and better explanations, but that covers ...
0
votes
0answers
21 views

Is the unit circle uniquely defined by all it's sliced averages?

Suppose you have a unit circle $g(x,y) = x^2+y^2-1 = 0$ and for each $\theta \in (-\pi/2, \pi/2)$ you associate a map $f_\theta(c): R \to R^2$ defined by $$f_\theta(c) = \langle g^{-1}(0) \cap ...
3
votes
2answers
63 views

Equation for Tangent Line that passes through $(0,1)$ on the curve $y = \ln x$

I'm totally lost. I've been trying to figure this out. This is what I've figured out: $dy/dx = 1/x$ $y$-intercept $= 1$ So I try to do $y-y_1 = m(x-x_1)+b,$ which I get as $y-1 = 1/x(x-0)+1,$ ...
0
votes
2answers
34 views

Finding the length from an interior point of a triangle to a vertex given distances to the other two

So let's assume that there is a triangle ABC and there is a point P inside of ABC. You are given the distances of AP and BP and you are trying to solve for CP. I faintly remember reading something ...
3
votes
4answers
80 views

Find the equation of a circle, given a point on it and a point where it is tangent to a given line

The given question is: Find the equation of the circle that passes through point $(-3,-4)$ and touches the line $x-y+7=0$ at the point $(-5,2).$ What I did was: Took the given points $(-5,2)$ and ...
-2
votes
1answer
15 views

What is the domain of the given function with the greatest integer?

The domain of the function $$f(x)=\sqrt{\frac{4-x^2}{[x]+2}}$$ where $[x]$ represents the greatest integer function, is (a) $(-\infty,-1)\cup[-1,2]$ (b) ...
0
votes
0answers
30 views

Arc measures in a circle

Suppose we have a quadrilateral inscribed in a circle prove that angles inside the same arc are equal
4
votes
1answer
97 views

A parabola lemma

I am looking for a previous reference and/or a geometric proof of the following lemma: Let $P$ be the parabola $y=x^2$. Let $a$, $b$, $c$, $d$ be four points on $P$ sorted from left to right, and let ...
-2
votes
2answers
36 views

Height of point on line in space [closed]

I have the coordinate $X$ and $Z$ of a point $M$. I need the $Y$ coordinate of this point knowing that the point is on a line defined in space by $A(X_1,Y_1,Z_1)$ and $B(X_2,Y_2,Z_2)$. Thank you
2
votes
1answer
29 views

Line with predefined length tangent to circle

I have one math problem which I'm trying to solve. I know it could be done but I'm a little bit "rusty" with my algebra. I'm kindly asking for help. Problem and procedure of my solution are shown in ...
0
votes
1answer
16 views

Find the vectorial equation of the line through $P$ and orthogonal to two planes

I have to find the vectorial equation of the line through $P$ and orthogonal to $r:(x,y,z)=(1,−1,−1)+\lambda(1,−1,0)$ and $s:(x,y,z)=(\frac{3}{2},-\frac{1}{2},0)+\alpha(\frac{1}{2},\frac{1}{2},1)$. ...
0
votes
2answers
25 views

Compute position of next point on a line

I'm writing a program in which it is possible to draw a horizontal, vertical or an oblique line. So the line can be described as follows : $f(x) = y = mx + q$ But my problem is that given the first ...
2
votes
5answers
92 views

Area of triangle bounded by line and degenerate “crossed lines” conic

The question is Show that the two lines given by $$(A^2 - 3B^2)x^2 + 8ABxy +(B^2 - 3A^2)y^2=0$$ and the line given by $$Ax+By+C=0$$ determine an equilateral triangle of area ...
1
vote
1answer
23 views

How to find the curve and axis?

We consider the following one-sheeted hyperboloid: $y^2-4x^2+4z^2=4$ This is also a surface (or solid) of revolution. So it must be generated by rotating a curve about an axis. What curve and axis ...
1
vote
1answer
39 views

How can I multiply by time?

I'm reading this article about collision detection. In it, he says: However, t appears to be referring to time - The time of ...
0
votes
1answer
17 views

Derive the 2-D analogue of the Laplace Dispersal Kernel using RDE

I found an interesting problem. I'm looking at the Laplace Dispersal Kernel for 1 dimensional dispersal behavior. And I wonder what happens in two dimensional world? I managed to find the limiting ...
0
votes
0answers
47 views

Analytical solution to nonlinear ode

I solved this equation that I attached numerically in matlab by the Newton Raphson method. Now I want to solve it analytically in matlab or even in Maple if it is possible. Would you please help me ...
0
votes
2answers
55 views

Linear algebra - find all possible positions of the third corner?

An equilateral triangle lies in the plane $x + y - z = 1$ and corners in points $(1, 1, 1)$ and $(2, 1, 2)$. Determine all possible positions of the third corner?
6
votes
4answers
110 views

finding the max of $f(x)=\sqrt{(x^2-4)^2+(x-5)^2}-\sqrt{(x^2-2)^2+(x-1)^2}$

I need to find the max of $$f(x)=\sqrt{(x^2-4)^2+(x-5)^2}-\sqrt{(x^2-2)^2+(x-1)^2}$$ When $x$ is a real number. What i did is to simplify: $$f(x)=\sqrt{x^4-7x^2-10x+41}-\sqrt{x^4-3x^2-2x+5}$$. Then ...
2
votes
2answers
81 views

Area of ellipse not in xy-plane

I've got a problem in which I'm trying to find the area of an ellipse which is given by the intersection of an elliptic cylinder with a plane. Nothing here is parallel to the coordinate axes, which is ...
2
votes
0answers
32 views

ShaderShop & ColorMaps [closed]

I'm trying to reimplement this: http://tobyschachman.com/Shadershop/ from scratch, just for fun. I have what I think is a good understanding up to where he transitions into the color map and 3 ...
0
votes
1answer
38 views

Finding equation of ellipse with given point and distance between directrices

I need to find the equation of an ellipse. The given were just a point where it passes, and distance between directrices. I know that the distance between directrices is given by $2a/e$. I don't ...
0
votes
1answer
27 views

Find the equation of a hyberboloid with given base, narrowest section, and the distance between them

I have one question left in an assignment and I havn't been able to solve it. I know the equaton for a hyperboloid and I know that $a$ and $b$ will be equal to each other. I don't know how to solve ...
1
vote
3answers
27 views

Finding circle of a sphere through two points

We have two points $P_1, P_2$ on a sphere $S$ of radius $R$. Suppose for $r \ll R$, the distance between $P_1$ and $P_2$ is less than $2r$. Then, $P_1$ and $P_2$ both lie on exactly two radius-$r$ ...
1
vote
0answers
40 views

Question about differential geometry [closed]

let $S$ be a subset of $\mathbb{R}^n$. We say that $S$ is path-connected if for each $x,y\in S$ there exists a continuous map $\gamma\colon[0,1]\to S$ (so we require $\gamma(t)\in S$ for all ...
1
vote
1answer
54 views

Intersecting two parabolas and computing the angle between the tangents in a point of intersection

I was solving some problems on parabola. I saw a question and solved it, but my solution was way too big. The question was: If $$\left(\frac{a}{b}\right)^{1/3}+\left(\frac{b}{a}\right)^{1/3} = ...
1
vote
0answers
41 views

Proof 5 points determine a conic without projective geometry

So I'm trying to prove that any five points, of which no 3 are colinear, there is a single conic that passes through al of them. I don't want to use projective geometry but rather, only analytic ...
1
vote
4answers
76 views

How to tell whether a point is to the right or left side of a line

I have a line equation in the form ax+by+c=0 and a point p(x,y).How can I determine on which side of the line the point is located?
1
vote
0answers
76 views

Calculus III problems [closed]

This problem takes place in a world with coordinates $x$, $y$, and $z$. An ant is running in a spiral on a plane that is floating down a river. The plane has its own polar coordinate system, and the ...
0
votes
1answer
21 views

Graphing and finding sine wave info with $\sin(x/4)$

I'd doing a chapter on graphing sine waves, and finding the amplitude, period, and so on. I know something like $y = 2 \sin(3x+ \pi) + 1$ can be turned into $y = 2 \sin[3(x + \pi/3)] + 1$ following ...