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

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3
votes
1answer
75 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" ...
5
votes
2answers
121 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
18 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
66 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
38 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
26 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
54 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: $$(y−y_0)=m(x−x_0)$$ $$(y+3/5)=m(x−1/2)$$ ...
0
votes
1answer
42 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 ...
3
votes
2answers
80 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
37 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 ...
3
votes
3answers
122 views

Property of ellipses involving normals at the endpoints of a focal chord and the midpoint of that chord

While solving a book on ellipses, I came across the following property of an ellipse which was given without proof :- If the normals be drawn at the extremities of a focal chord of an ellipse, a ...
0
votes
2answers
44 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
65 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
42 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
133 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 ...
0
votes
0answers
65 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
34 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
145 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
81 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 ...
4
votes
4answers
112 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 ...
-3
votes
1answer
18 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
34 views

Arc measures in a circle

Suppose we have a quadrilateral inscribed in a circle prove that angles inside the same arc are equal
7
votes
1answer
121 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
1answer
32 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
19 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
41 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
4answers
122 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
28 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
42 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
24 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
2answers
56 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
111 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
107 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 ...
0
votes
1answer
69 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
34 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
34 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
1answer
84 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
72 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
218 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?
0
votes
1answer
24 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 ...
0
votes
0answers
38 views

Prove $(\vec{r}\cdot\nabla)\vec{u}+\vec{r}\times (\nabla \times \vec{u})=\vec{r}(\nabla\cdot\vec{u})+(\vec{r}\times \nabla )\times\vec{u}$

Show that $$(\vec{r}\cdot\nabla)\vec{u}+\vec{r}\times (\nabla \times \vec{u})=\vec{r}(\nabla\cdot\vec{u})+(\vec{r}\times \nabla )\times\vec{u}$$ I have been trying to show this for the past few ...
0
votes
2answers
51 views

How to connect a line between 4 randomly placed points on a plane such that the line does not cross itself

You get 4 coordinates of points on a plain. You need to connect them all with a line. The line must not cross itself. What's your strategy?
0
votes
1answer
38 views

Equation of a parabola: Translations and rotation

I've tried to solve this problem: Find an equation of the parabola with vertex at point $(1,1)$ whose directrix is the line $x-2y=6$. It has to be solved using translation and rotation (coordinate ...
1
vote
1answer
52 views

Find the distance of a point from a plane generated by two given vectors

I need to calculate the distance of the point $P = (0, 5, -4)$ from the plane which pass from the point $P1=0, 1, -2)$ and generated by the two vectors: $$ v1 = (1, 2, 3), v2 = (-1, \sqrt{2}, 1) $$ ...
0
votes
0answers
69 views

What is the best book to learn coordinate geometry

The level should be above high school, and it must be free online if at all possible. Also, I have an additional question to ask: How many months, roughly, would it take to finish a mathematics book ...
0
votes
2answers
78 views

What is the minimum information required to define an equation for ellipse?

What is the minimum information ie. amount of points in 2-dimensional plane in order to define the equation for an ellipse? I know that unique ellipse cannot be defined when only one of the foci is ...
0
votes
1answer
41 views

What is the best book for coordinate geometry?

Requirements: A) not too thick , as I am reading this only to solve calculus problem . B) free on web is the best (optional) C) I don't mind if the book involves both coordinate geometry and ...
1
vote
2answers
56 views

Can ellipse equation be transformed through one of its foci?

Can we transform ellipse equation to represent an ellipse transformed by tilting it through its focus such that its center point moves in circular manner and one of its focus stays at constant ...