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

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3
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3answers
145 views

What equation intersects only once with $f(x)=\sqrt{1-(x-2)^2}$

Being $f(x)=\sqrt{1-(x-2)^2}$ I have to know what linear equation only touches the circle once(only one intersection), and passes by $P(0,0)$. So the linear equation must be $y=mx$ because $n=0$. I ...
1
vote
2answers
640 views

Finding point coordinates of a perpendicular

Given that I know the point coordinates of point $A$ and point $B$ on segment $AB$ and the expected length of a perpendicular segment $CA$ (perpendicular over $AB$), how do I calculate the point ...
0
votes
1answer
205 views

Which surface is formed by rotating a hyperbola around its asymptotes?

I don't know even what a type of surface will be. And what equation will be? The equation of hyperbola - $$ xy = l. $$ Now, let's $$ x = x'cos(\varphi ) - y'sin(\varphi ), y = x'sin(\varphi ) + ...
1
vote
0answers
98 views

A function with the same slope as $b\sqrt{\frac{x^2}{a^2}-1}$ but not imaginary in [0,a]?

For some fixed $a,b \in \mathbb{R}$, $y = b\sqrt{\frac{x^2}{a^2}-1}$ is supposed to plot the boundary of an ellipse in $\left[0,a\right]$. I came up with that function but it has the defect that it ...
2
votes
2answers
1k views

How to calculate distance between point and object in 3d space

I have object in 3d space created from points $P_i(x, y, z)$ from which I can create triangles, and I need to calulate distance from point X to this object. I try to take 3 points from smallest ...
0
votes
2answers
400 views

Analytic Geometry in Space

Can someone help me solve the following two questions: 1) Find the distance between the lines: $$ L_1: \frac{x-1}{2} = \frac{y+3}{1} = \frac{z}{-1}$$ and $$\displaystyle L_2 : ...
1
vote
1answer
93 views

Is the study of algebraic curve is techniquely equal to the advanced division of analytic geometry, if not, what is the difference?

Is the study of algebraic curve is techniquely equal to the advanced division of analytic geometry, if not, what is the difference? And what is other branch of advanced analytic geometry called? in ...
0
votes
2answers
187 views

Find the parallels to a line which are tangent to an ellipse

Having the equation of a line, how can I find which of its parallels are tangent to an ellipse of equation $x^2 + 9y^2 = 1$? If the equation of the line is $y = mx + q$, I know that its parallels ...
2
votes
1answer
1k views

Finding a vertex of a triangle knowing the other two and its area

I have vertix A, vertix B and the area of a triangle, and I need to find the coordinates of vertex C, knowing that it's on the bisector between the first and the third sector of the Cartesian plane. ...
1
vote
1answer
2k views

Parametric equation for a line which lies on a plane

Struggling to begin answering the following question: Let $L$ be the line given by $x = 3-t, y= 2+t, z = -4+2t$. $L$ intersects the plane $3x-2y+z=1$ at the point $P = (3,2,-4)$. Find parametric ...
1
vote
2answers
98 views

Rectangle area and a curve

The diagonals of a rectangle are both 10 and intersect at (0,0). Calculate the area of this rectangle, knowing that all of its vertices belong to the curve $y=\frac{12}{x}$. At first I thought it ...
1
vote
0answers
349 views

Prove a very original version of Descartes's circle theorem

Prove: I define the radius of three mutually externally tangent to be $d,e,f$ respectively. The circle with radius $x$ is internally tangent to all three circles. Then $$ddeeff+ddeexx+ddffxx+eeffxx ...
1
vote
3answers
1k views

Find unknown coordinates of points

I hope it's enough understandable.
3
votes
1answer
110 views

Determinant and Measure

The determinant of the matrix of its vectors gives the measure of an $n$-dimensional parallelogram. For example, in $2$ dimensions, the area spanned by vectors $v$ and $w$ is \begin{array}{|cc|} v_1 ...
1
vote
2answers
280 views

Can more than four circles internally tangent or external tangent or combination of both each others at different points?

Is it true for infinite number of m, more than four, there exist m circles internally tangent or external tangent or combination of both each others(in this problem, i mean a circle must be tangent to ...
1
vote
1answer
376 views

closest point on a plane to another point in $\mathbb{R}^3$

Given $4$ points in $\mathbb{R}^3$: $A(0,2,4);B(-2,6,-2);C(2,-4,8);D(10,2,0)$, find the line equation $AK$ when $K$ is the projection of $D$ on the plane $ABC$. The first thing I did was find the ...
1
vote
1answer
158 views

Disk integration method to find volume of solid of revolution

I know that in a classic Cartesian coordinate system $xOy$, if I have a function $y = f(x)$ and I want to find the volume of the a solid of revolution around x-axis I can compute: $$V = \pi ...
2
votes
2answers
56 views

Are these planes?

Given the equation: y + z = 10 Can it be considered a plane? Why (not)? How do you correctly express planes which are normal onto one axis, for example a plane that lies completely vertical in space ...
0
votes
1answer
110 views

What is the formula of the following?

Let $S$ be the ellipsoid given by the formula $$ \frac{x^2}{a^2}+\frac{y^2}{b^2} +\frac{z^2}{c^2}=1$$ where $a \ge b \ge c > 0$ are fixed constants. What is the formula given by the set consisting ...
0
votes
1answer
38 views

Find a circumference with center on a line

I have a set of circumferences $$x^2 + y^2 + \alpha_1 x + \beta_1 y + \gamma_1 + k(x^2 + y^2 + \alpha_2 x + \beta_2 y + \gamma_2) = 0$$ $\alpha_1, \alpha_1, \beta_1, \beta_2, \gamma_1, \gamma_2$ ...
4
votes
3answers
3k views

Find a point on a line segment, located at a distance $d$ from one endpoint

Given points $A$ and $C$ in the plane, how do I find the point $B$ on the line segment between $A$ and $C$ that is located at a distance $d$ from $A$? Example: $$A = (0,3), \qquad C = (3,0), \qquad ...
1
vote
2answers
550 views

Plane from intersection line and point

The task: Determine the plane containing point $P( -5 , 2 , 3 )$ and going through the intersection line of the planes $2x + y + 5z = 31$ and $-4x + 5y + 4z = 50$ 1.: Intersect the two given planes, ...
0
votes
1answer
132 views

Equations in Analytic Geometry

There are many equations in Analytic Geometry like equation of a line, equation of a plane etc. My question: 1) Why equations instead of functions? 2) Why do equations almost always equal zero?
4
votes
1answer
153 views

Is there any way to give sense to a geometric/visual proof?

Suppose one is given the following visual proof that $$\lim\limits_{n \to \infty} \sum_{k=1}^n \frac{1}{2^k} = 1$$ which is the following construction over $[0,1]\times[0,1]$ What this is ...
1
vote
1answer
277 views

What do the parameters skewX and skewY mean in the transform specified by Flash's motion XML?

Flash has the ability to export animations into a format they call motion XML. Its specification is here I am trying to write a python renderer for these animations using pyglet. I understand ...
1
vote
1answer
121 views

Directional cosines of a line.

Show that if the lines with the directional cosines $(l, m, 0)$ and $(p, 0, q)$ are perpendicular then either $m = \frac {1}{\sqrt{p^2 + q^2}}$ or $q = \frac {1}{\sqrt {l^2 + m^2}}$.
0
votes
1answer
79 views

One spot and distance known, Second spot unknown

I know the coordinates of points E and Q, so I know their euclidean distance L. I'm looking for the point W with coordinates (a,b) related to other known values?
4
votes
3answers
542 views

What is the tangent plane equation on the 3 spheres?

3 spheres are on $z=0$ plane and touch each other as shown in the picture. Coordinates of their centers are $O_1=(0,0,5),O_2=(0,y_2,3),O_3=(x_3,y_3,2)$. What is the tangent plane equation on 3 ...
4
votes
3answers
250 views

How do I mathematically halve Orange Juice with my brother!

I need to calculate the height of a glass(frustum) where the volume is half of total volume. Obviously, at h/2, volume will not be v/2. So my question is, at what height from the bottom of the glass ...
0
votes
2answers
237 views

Help me with Cylinder -coordinates problem, back to Cartesian or not? How to do it fast?

Source of the problem, 3b here. Problem Question Electricity density in cylinder coordinates is $\bar{J}=e^{-r^2}\bar{e}_z$. Current creates magnetic field of the form ...
1
vote
0answers
62 views

geometry of points in $\mathbf{Z^3} $ and center of mass

Given the set of $37$ points in $\mathbf{Z^3} $ in which no 4 points are on the same plane. Show that there exist 3 point A,B,C in this set such that $ (\frac{x_A+x_B+x_C}{3}, ...
2
votes
3answers
472 views

How to fill up the gap between a typical advanced undergraduate algebraic curve course and High school basic geometry/precalculus course?

Based on this question i asked recently: A question about geometry of plane curve books, i think it is too advance for a HS student/ typical second or third year undergraduate math majors to read on ...
3
votes
2answers
1k views

How to calculate angles and X,Y coordinates for drawing a hand of playing cards on a canvas

Scenario: I'm programming a module to draw a deck of cards on a canvas as though they were being held by a person. Edit: I've cleaned up the question as best I can to be clearer. What I'm looking ...
1
vote
0answers
2k views

Direction ratio of a vector in 3d?

Suppose I have a vector whose starting end is the origin & the other end can be in any of the 8 quadrants (in 3d). I can easily get direction ratios for the vector & also know in which ...
0
votes
1answer
138 views

“Way” to decide if points are in a rectangle.

Suppose $P_1=(x_1,y_1)$, $P_2=(x_2,y_2)$ are two points. Also suppose that we have a rectangle which we just know the value of its sides $a$ and $b$. I am looking for some kind of formulation which ...
2
votes
0answers
44 views

elipsoid surface intersection in $\mathbb{R}^3$

Is there an explicit parametric solution describing the curve result of the intersection of two elipsoid surfaces with abitrary position and orientation in $\mathbb{R}^3$?
0
votes
1answer
34 views

Not very clear about “parametric form”

For example, let the surface $S$ in $\mathbb R^3$ be formed by taking the union of the straight lines joining pairs of points on the lines $$\{x=t,y=0,z=1\},\qquad \{x=0,y=1,z=t\}$$ with the same ...
1
vote
3answers
386 views

Constructing a line with a known line, intersection point and angle.

I am creating a Java game with collisions. I found myself stuck on the following problem. I have got two known lines: $y$ and $i.$ $i$ is the inbound direction and $o$ the outbound direction, ...
0
votes
1answer
2k views

Given two vertices, how to find the other two vertices of a rhombus?

$A\;(-3,-4) $ and $ C \; (5,4)$ are the ends of the diagonal of a rhombus $ABCD$. Given that the side BC has gradient $\frac{5}{3}$; How could we find the coordinates of $B$ and hence of $D$? ...
2
votes
1answer
311 views

Slope of a line perpendicular to a line of slope $0$.

I have three points: $P = (2,5)$ $Q = (12,5)$ $R = (8,-7)$ I need to find the equation of the line thru $R$ which is perpendicular to $PQ$. How do I do this? $PQ$'s gradient is ...
2
votes
1answer
49 views

Finding unknown 3D vector given 2 known vectors and 2 known angles

So I have a 3D vector math problem that I'm having difficulty solving. Basically I have two known vectors in the form (x,y,z), let's call them C and P, and I want to find a third unknown vector, let's ...
5
votes
1answer
368 views

What is the path equation that is created with the middle point of a fixed length line segment that touching both ends to an ellipse.

Ellipse equation is $(\frac{x}{a})^2+(\frac{y}{b})^2=1$ and the length of line segment is $2k$, if we move the line segment all around of the ellipse while touching both ends to the ellipse. What is ...
-1
votes
1answer
106 views

Properties of lines in the Pixel + Zoom geometry

Problems Prove that in PZ geometry, every PZ line has an equation of the form ax+by=c, where a, b, c are all rational numbers and a and b are not both zero. Prove that every equation of the form ...
11
votes
5answers
3k views

Calculating the area of an irregular polygon

Given the length of the sides of an irregular polygon (no coordinates provided) how do you compute the area of the maximum area of the polygon? Thanks in advance
1
vote
1answer
403 views

Finding the coordinates of a point

I have the $x$ and $y$ values to $2$ points $A(x_a,y_a)$ and $B(x_b,y_b)$. I need to determine the coordinates of a third point $S(x_s,y_s)$ that is at $r$ distance away from both A and B points and ...
0
votes
2answers
89 views

Existence of n-dimensional polyhedron given edges

The following assertion is true in $2$ and $3$ dimensions: Given $\sigma_{ij},\ 1\leq i\neq j\leq n$ with $\sigma_{ij}=\sigma_{ji}$ and $\sigma_{ij} \leq \sigma_{ik}+\sigma_{kj}$, then there exist ...
0
votes
2answers
2k views

Three-circle intersection for circles of unbounded integer radius

I have three circles. One is at $(0,0)$ and has radius $n$, another has is at $(1,0)$ and has a radius $m$, and the third is at $(0.5, \sqrt{0.75}))$ and has a radius of $o$. All of the radius values ...
3
votes
4answers
3k views

How to calculate the two tangent points to a circle with radius R from two lines given by three points

I need to calculate the two tangent points of a circle with the radius $r$ and two lines given by three points $Q(x_0,y_0)$, $P(x_1,y_1)$ and $R(x_2,y_2)$. Sketch would explain the problem more. I ...
2
votes
2answers
782 views

Problem Solving Question Relating to Directions and finding Burger Jack

I stopped at a street corner and asked for directions to Burger Jack. Unfortunately, the person I wasked was Larry Longway, whose directions are guaranteed to be too complicated. He said,"You are now ...
0
votes
2answers
1k views

Perpendicular Vectors

Find the equation of the line passing through a point $B$, with position vector $ \vec b$ relative to an origin $O$, which is perpendicular to and intersects the line $\vec r= a+ \lambda \cdot c$, ...