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

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Equation of common tangent(s) to two parabolas

Suppose we are given two parabolas, described by their directices $d_1: a_1x+b_1y+c_1=0, d_2: a_2x+b_2y+c=0$ and foci $F_1(p_1,q_1), F_2(p_2,q_2)$. How does one find equations of common tangents to ...
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3answers
138 views

How can the centers of these 5 related circles be specified as a formula?

This is my first time posting in this forum, so please forgive me if my question is too involved or if I've posted it in the wrong area. I hope someone finds it interesting enough to try their hand at ...
11
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4answers
254 views

Tricky 3d geometry problem

We have a cube with edge length $L$, now rotate it around its major diagonal (a complete turn, that is to say, the angle is 360 degrees), which object are we gonna get? Astoundingly the answer is D. ...
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1answer
78 views

How to get coordinates of a point after an image is rotated? (with images)

I have a problem that involves a rotating image and finding a previously known point. Firstly, there is a sequence with the rotation. We start with an empty image. A line is drawn vertically, ...
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3answers
553 views

Find the coordinate of third point of equilateral triangle.

I have two points A and B whose coordinates are $(3,4)$ and $(-2,3)$ The third point is C. We need to calculate its coordinates. I think there will be two possible answers, as the point C could be on ...
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2answers
64 views

Finding the coordinates of a parallel line given line coordinates and a distance

I have a path defined by a list of (x, y) coordinates and I want to create two additional paths, one offset by a distance of 0.25, the other by -0.25. I think that could be done by finding parallel ...
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1answer
17 views

Find the equation of the hyperbola that satisfies this condition

Focus is at $F\equiv(−3−3√13, 1)$, asymptotes intersect at the point with coordinates $(−3, 1)$ and one asymptote passes through $(1, 7)$ I've solved some problems that involve equations of ...
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1answer
234 views

Simplifying an equation (circle)

I'm trying to work on a problem, and I'm stuck at simplifying this equation. I do not why I cannot see it: so the book gives the following equation: $$\frac{ax}{x^2+y^2}+\frac{by}{x^2+y^2}+c=0$$ ...
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1answer
40 views

Why 9 points determine a quadric

The books I have state this redult as obvious from the definitions, but it is not clear to me why this is so.
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32 views

Find the direction from which the projected area of a loop is maximal

How do I find the direction from which the projected area of a loop is maximum? Should I try to use intuition or is there a simple mathematical way to find it? The problem given was the following: ...
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1answer
22 views

How do I find the symmetrical point B given the centre of symmetry C and another point A?

I have a point $A (-2k; 3)$ and a point $B$ that is symmetrical to the point A given the centre of symmetry $C (-1; 0)$. I tried applying the following formula, where $x_o$ and $y_0$ are the ...
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61 views

A problem about the intersection of a plane and a sphere

So I have been trying to solve this (rather basic) geometry problem but don't know how to parametrize the functions and get an answer (please check below). Ok, so my problem reads like this: prove ...
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1answer
89 views

Given one endpoint and midpoint in (x,y) of a line segment, explain how to find the other end point.

A line segment with one end at C(6,5)has midpoint M(4,2). Determine the coordinates of the other endpoint, D. Explain your solution and describe a method to check your answer.
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6answers
217 views

Area enclosed by the graph of $13x^2-20xy+52y^2+52y-10x=563$.

Find the area enclosed by the graph of $13x^2-20xy+52y^2+52y-10x=563$. First I saw that this cannot be a circle ($xy$ term), and it cannot be an ellipse with axes parallel to the coordinate axes. But ...
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6answers
94 views

Why is the equations for a perpendicular line $-\frac{1}{m}$?

Why is it just $-m$? Lets say $y=x$ and the $y$ intercept is at $0$ if we created another line that was $y=-x$, wouldn't that make it perpendicular? Note: Don't exactly know what tags to use, feel ...
3
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1answer
76 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" ...
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2answers
125 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 ...
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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, ...
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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?
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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…: ...
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1answer
28 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 ...
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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: $$(y−y_0)=m(x−x_0)$$ $$(y+3/5)=m(x−1/2)$$ ...
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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 ...
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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 ...
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1answer
38 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 ...
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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 ...
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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 $m$ ...
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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 ...
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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 ...
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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?
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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) = ...
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1answer
137 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 ...
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68 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 ...
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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 ...
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2answers
151 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 $y-1 = 1/x(x-0)+1,$ ...
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2answers
84 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 ...
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4answers
113 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 ...
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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) ...
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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
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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 ...
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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 ...
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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 $s:(x,y,z)=(\frac{3}{2},-\frac{1}{2},0)+\alpha(\frac{1}{2},\frac{1}{2},1)$. ...
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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 ...
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4answers
123 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 ...
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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 ...
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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 ...
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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 ...
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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?
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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
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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 ...