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

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3
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1answer
82 views

reference on $\sqrt{ax}+\sqrt{by}=c$ as a parabola?

Does anyone have a reference on the equation $$\sqrt{ax}\,+\sqrt{by}=c\ ?$$ Clearing square roots and rearranging gives $$ax+by = \frac{(ax-by)^2+c^4}{2c^2}$$ This is the equation of a parabola, so ...
0
votes
1answer
44 views

differentiating an integral with respect to a variable which also affects the region of integration

I am considering taking the derivative of the function $$F(\mathbb{x_1},\mathbb{x_2},\mathbb{x_3}) = \displaystyle \int_{V_1} ||x-\mathbb{x_1}||\phi(x)\,dx + \int_{V_2} ||x-\mathbb{x_2}||\phi(x)\,dx ...
2
votes
3answers
64 views

Drawing circumference issue

I'm a developer, and I'm developing an app on Google Maps. At the moment, I'm trying to draw a circle on the map. For getting all the points I need, I'm using the following formula: \begin{equation} ...
1
vote
1answer
33 views

Interpolating Random Points

I have a list of (x,y) co-ordinates that need to be interpolated. The co-ordinates are not necessarily part of a function. Therefore, polynomial interpolation will not work. Is there a way to use some ...
3
votes
1answer
65 views

Rotation of complex numbers in a complex plane. Check my work?

Say that $c_1 = -i$ and $c_2 = 3$. For this problem, let $z_0$ be an arbitrary complex number. We can rotate $z_0$ around $c_1$ by $\pi/4$ counterclockwise to get $z_1$. Next, we canrotate $z_1$ ...
2
votes
4answers
125 views

how to prove that the circle $(x-a)^2+(y-b)^2=a^2+b^2$ is passing through point $(0,0)$

How can one prove that the circle $(x-a)^2+(y-b)^2=a^2+b^2$ is passing through point $(0,0)$? I know that if i put: $x=y=0$, i will get: $(0-a)^2+(0-b)=a^2+b^2=a^2+b^2$ But that's not a proof but ...
1
vote
4answers
118 views

Median of triangle

I know that a median of a triangle is a line joining one of the vertices to the mid-point of the opposite side. For example, in a triangle OAB, O is the origin, $A$ is the point $(0,6)$ and $B$ is ...
0
votes
2answers
46 views

Find value of $t$ between the difference of 3D vectors.

Hint: The distance between $2$ vectors equals the magnitude of their difference. What is the value of $t$ for which the vector $\mathbf v = \begin{pmatrix} 2 \\ -3 \\ -3 \end{pmatrix} + ...
3
votes
2answers
64 views

Distance involving 3D lines and vectors.

In this problem, a = \begin{pmatrix} 5 \\ -3 \\ -4 \end{pmatrix} and b = \begin{pmatrix} -11 \\ 1 \\ 28 \end{pmatrix} Vectors p and d exist such that the line containing a and b can be expressed in ...
0
votes
1answer
48 views

Determining positions of straight line

I need to find out the relative positions to each other a straight line, first I'm trying to check if they are coplanar but I get an unknown variable. Can anyone help me on how to solve this part of ...
1
vote
2answers
32 views

3D line in a 3D plane. Find the intersection of the two.

(I'm new to Math.StackExchange, so if you see any errors, please comment below!) $\mathcal{P}$ is the plane containing the three points $(-3,4,-2)$, $(1,4,0)$, and $(3,2,-1)$. $\ell$ is the line ...
0
votes
0answers
28 views

Lines on a specific cubic surface

Consider the cubic surface given in affine coordinates by the equation $x^2+y^2=g_3(z)$ ($g$ is a polynomial of degree 3 s.t. the cubic is smooth). Is it possible to write down explicitly the ...
0
votes
2answers
35 views

Find a bisector point of a circle

The coordinates of $A=(x_{0},y_{0}$) and $B=(x_{1},y_{1}$) are given. How to find the coordinates of $C$ and $D$ as per given information below. ABC is equilateral triangle such that $AB=BC=CA$ ...
0
votes
0answers
36 views

What's the area of the shape defined by all points whose distances from two focal points multiply to give the same product?

This shape, which I call the multiplicoid, is the equivalent of, and very similar to, an ellipse. However, instead of the distance between each point and the two focal points summing to a constant, ...
8
votes
4answers
133 views

Is it possible to put an equilateral triangle onto a square grid so that all the vertices are in corners?

In the following collection of problems - arXiv:1110.1556v2 [math.HO] - the following question is posed: Is it possible to put an equilateral triangle onto a square grid so that all the vertices ...
0
votes
2answers
14 views

Centering Text on Image after padding values

I'm creating an Image, the Image has text on it. The text I want it to be center on the Image but the Image has padding values that I assigned. The equation I'm using and the values for the ...
0
votes
1answer
25 views

Coordinates of regular octagon

There are coordinates of two vertices in a regular octagon $$\{A_0,A_1,...,A_7\}$$ with Cartesian coordinates $A_0 = (2,-4)$, $A_2 = (0,0)$. The task is to find coordinates of all other vertices. ...
-3
votes
1answer
44 views

Find a plane that passes through a point $p$ but is perpendicular to a line $\ell$ [closed]

The point $p$ that I am using is $(1,6,-4)$ and line $\ell$ is $$ \begin{cases} x = 1 + 2t\\ y = 2 - 3t \\ z = 3 - t \end{cases} $$ Other points for the line I have found are $(1,2,3)$ and ...
-3
votes
3answers
65 views

is there an online tool for solving equation of a line? [closed]

I have as input two points. But the input points contain variables (constant references). I've seen some tools online but they require numeric values. The two given points are specified by constant ...
-1
votes
2answers
49 views

How can one calculate the limit of $\frac{1}{x^2-9}$ as x approaches -3 and 3 by hand? [closed]

Reviewing math for college after a gap year and so I know this is probably a pretty elementary question, but let me know if it has any interesting implications or alternative solutions or if it ...
0
votes
1answer
54 views

Intersection of two lines in complex numbers given four points [closed]

How to find the point of intersection of two lines, given four points, two of which are on each line, in complex numbers? Thank you!
0
votes
0answers
16 views

An integral from the integral geometry about the isoperimetric inequality.

The problem is from the book "Integral Geometry and Geometric Probability" by Santalo (1976), Chapter 1.3.5, Notes and Exercises (page 37). Given a convex closed curve $C$. Let $A_1$, $A_2$ be the ...
2
votes
1answer
23 views

Strange “form” of the set of vertices $C(x,y,z)$ such that $ABC$ is a right triangle with hypotenuse $AB$

Let $A(1,-3,4)$ and $B(3,-2,-1)$ and find the set of all $C(x,y,z)$ such that $ABC$ is a right triangle with hypotenuse $AB$ What I did $$AB=(2,1,5)$$ $$BC=(x-3,y+2,z+1)$$ $$AC=(x-1,y+3,z-4)$$ ...
2
votes
4answers
118 views

Integrating $\sqrt{1-x^2}$ without using trigonometry

I am a beginning calculus student. Tonight I had a thought. Maybe I could calculate $\pi$ using integration, but no trig. The problem is that I don't really know where to start. I thought perhaps I ...
-1
votes
2answers
41 views

Given the endpoints of a line segment, develop the equation of its perpendicular-bisector

Find the equation of the perpendicular bisector of $AB$ for: $A(1\mid 3)$ and $B(-3\mid 5)$. What I did: $m=\frac{3-5}{1+3}=-\frac12$ for the slope of $AB$ $(\frac{3+5}2\mid\frac{1-3}2)=(4\mid -1)$ ...
0
votes
2answers
42 views

Deduce the inequalities $3\lt \pi \lt 12(2-\sqrt{3})$, by calculating the areas of regular twelve-sided polygons.

Calculate the areas of regular dodecagons (twelve-sided polygons) inscribed and circumscribed about a unit circular disk and thereby deduce the inequalities $3\lt \pi \lt 12(2-\sqrt{3})$. This is a ...
2
votes
0answers
46 views

A better way to answer this question

So my team and i were asked this question a few years ago on a small Math-A-Thon on my hometown. It went something like this: "We need to transport a neon tube (or any tube, who cares) of 92cm ...
2
votes
2answers
44 views

Find the Ratio $BM \colon ME$

In Triangle $\Delta ABC$, the Point $D$ is on $BC$ such that $D$ divides $B$ and $C$ in the Ratio $1 \colon 3$ and there is a point $E$ on $CA$ such that $E$ divides $C$ and $A$ in ratio $1 \colon 3$. ...
0
votes
1answer
28 views

Equation of the locus

Find the equation of the locus of a point $P = (x, y)$ when the sum of the squares of the distances from $P$ to the points $(a, 0)$ and $(-a, 0)$ is $4b^2$, where $b \geq \dfrac{a}{\sqrt{2}}$?
3
votes
1answer
45 views

How to get rid of the term with $xy$?

I'm trying to put this conic on an identifiable form. $$4x^2-4xy+y^2+20x+40y=0$$ I know that the term $xy$ implies that I need to rotate the conic so that $xy$ vanishes. But I've read on some books ...
0
votes
0answers
22 views

Should the expanded expression of a quadratic form be equals to It's original expression?

Sorry if the question is a little misleading, but I have no better way to express it. The text below should clarify. Suppose I have the equation of a conic: $x^2+y^2+z^2-2x+3y+z+2=0$, with this I ...
1
vote
1answer
33 views

Point within a Cube in 3D environment

I have a cube in 3D space with 8 corner points with their X,Y,Z Coordinates. I know how to test if any given point lies inside a cube by Comparing their coordinates to be greater or smaller than the ...
4
votes
1answer
49 views

Start and end point of a rotated ellipse

I have the data of an incomplete ellipse and I need to retreive the minimun information in order to describe an elliptical arc. In particular following are my ellipse data: Major axis vector (x, y) ...
1
vote
2answers
41 views

Given a Line Parametrization, Finding another Equation

So I am given a line $l$ with the parameterization, $x=t, y=2t, z=3t$. Now let some point, $p$ be a plane that contains the line $l$ and the point $(2,2,2)$. So given this, how do I find an equation ...
2
votes
1answer
18 views

Where are these choices of $A',B',C'$ for this quadratic form?

I'm studying quadratic forms: In the book I'm reading, he starts by looking at quadratic forms such as: $$\varphi (x,y)=Ax^2+2Bxy+Cy^2$$ And that given this quadratic form, one can introduce via ...
3
votes
2answers
25 views

The Reason for different Forms of Equations

I recently started learning about conic sections and saw people writing the equations for the different figures (circle, parabola, ellipse, and hyperbola) in different forms. (standard form, vertex ...
1
vote
1answer
22 views

For line $ax+by+k=0$ which intercepts form a triangle rectangle with area $A$, find $k$

I know that the area of a triangle is given by the formula $A=\frac{1}2Bh$ and the intercepts of line $ax+bx+k=0$ are $(B,0)$ and $(0,h)$ which forms a square with area $2A$, but without brute-forcing ...
0
votes
1answer
40 views

When should I shift $a$ and $b$ in $\cfrac{x^2}{a^2}+\cfrac{y^2}{b^2}=1$?

Find the reduced equation of the elypsis such that: The foci are $(0,6);(0,-6)$ and the larger axis has length $34$. I did the following: Taking the equation ...
1
vote
0answers
27 views

Question about the coordinates in a new origin on the plane.

I'm reading a book on analytic geometry, specifically on a chapter on change of coordinates. It says that having the origin $O$, one point $P$ and a new origin $O'$, the vector that describes the ...
1
vote
1answer
31 views

A simplified formula for area of triangle when equations of the sides are given

For i = 1, 2, and 3, let $a_ix + b_iy + c_i = 0$ be three equations of 3 (non-special cased) straight lines. From which, the co-ordinates of the vertices can be found. Using these co-ordinates, via ...
1
vote
1answer
31 views

Why does this hyperboloid change into a surface? [duplicate]

Given this equation $x^2+y^2+z^2+2xy+2xz+2yz-x-y-z=6$ and the corresponding quadric: If I rearrange the equation to $(x+y+z-3)(x+y+z+2)=0$ (which is equivalent), I get: So, which is the right ...
1
vote
1answer
30 views

Points on two skew lines closest to one another

Given two skew lines defined by 2 points lying on them as $(\vec{x}_1,\vec{x}_2)$ and $(\vec{x}_3,\vec{x}_4)$. What are the vectors for the two points on the corrwsponding lines, distance between ...
3
votes
0answers
41 views

How to transform (rotate) this hyperbola?

Given this hyperbola $x_1^2-x_2^2=1$, how do I transform it into $y_1y_2=1$? When I factor the first equation I get $(x_1+x_2)(x_1-x_2)=1$, so I thought $y_1=(x_1+x_2)$ and $y_2=(x_1-x_2)$. ...
5
votes
4answers
163 views

Prove that a point can be found which is at the same distance from each of the four points$\ldots$

Prove that a point can be found which is at the same distance from each of the four points $\bigg(am_1,\dfrac{a}{m_1}\bigg),\bigg(am_2,\dfrac{a}{m_2}\bigg),\bigg(am_3,\dfrac{a}{m_3}\bigg)$ and ...
2
votes
2answers
31 views

Find the equation of base of Isosceles Traingle

Given the two Legs $AB$ and $AC$ of an Isosceles Traingle as $7x-y=3$ and $x-y+3=0$ Respectively. if area of $\Delta ABC$ is $5$ Square units, Find the Equation of the base $BC$ My Try: The ...
0
votes
1answer
28 views

Area of Triangle Given 3 vertices

Given that $P=(1,1,0), Q=(1,0,1), R=(0,1,1)$. I need to find the area of the triangle. What I have done: I have tried finding the distances of PQ, QR, and PR. I have those distances, I don't know ...
-2
votes
2answers
33 views

Co-ordinate geometry and area of triangle

When a straight line $ax+by+c=0$ forms a triangle with the axes $x$ and $y$, what is the general formula for the area of the triangle?
2
votes
1answer
58 views

What geometric object is given by this equation?

What geometric object is given by this equation? $x^2+y^2+z^2+2xy+2xz+2yz-x-y-z-6=0$ Maple says it's a hyperboloid of one sheet, but is there a way to show it without going the long way by using the ...
3
votes
2answers
90 views

Let $ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$. Determine $S+S+…+S $.

Let $$ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$$ By the usual notation for sum of sets let $$ 2S\overset{\text{not}}{=}S+S=\{(x_1+x_2,y_1+y_2) \ | \ (x_1,y_1), ...
6
votes
0answers
53 views

Is this solution legal?

Let $M(1,-1)$ be a point in a plane. Find its distance from a line given by $x+2y-4=0$. Later on I found a formula: $$d=\frac{\left | Ax_{0}+Bx_{0}+C \right | }{\sqrt{A^2+B^2}}$$ But I did it ...