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

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2answers
49 views

How to find whether a point lies on a line which is in parametric form?

Does the point $(1,8,3)$ line on the line with parametric equation: $$x = 5 + 2t$$ $$y = 2 + 6t$$ $$z = 1 + 3t$$ I know how to solve if they give me a equation of a plane and ask whether the ...
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1answer
13 views

A problem about affine spaces

Let A be an affine space,dim(A)=4. P,Q are planes from A. If dir(P)!=dir(Q),then P and Q are disjoint. Is this proposition true or false? I know that two planes are parallel if they are disjoint ...
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3answers
48 views

A question about an equation of a plane

Let $A=(1,3,1)$; $B=(1,1,1)$; $C=(2,0,1)$; $D=(1,-2,3)$. Determine the equation of a plane that passes through $D$ and is parallel with $(ABC)$. I know the fact that $\mbox{dir}(\text{plane})=\mbox{...
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1answer
21 views

Find the radius of the circle with some given conditions.

A circle having centre at C is made to pass through the point $P(1,2)$ , touching the straight lines $7x - y = 5$ and $x + y +13 = 0$ at A and B respectively. Then find the radius of the circle. I ...
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1answer
44 views

Polar coordinates in taxicab geometry

We know that in euclidean $\mathbb{R}^2$ space polar coordinates are defined by $$r = \sqrt{x^2 + y^2}$$ $$\theta = \arctan\frac{y}{x}\text{.}$$ Now, geometrically we can think of it as of point, ...
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1answer
31 views

Finding the point on the ellipse under certain conditions

This is a kind of simple question, but it gives me hard time: An ellipse is given in coordinate system. It passes points $(a, 0)$, $(0, b)$, $(-a, 0)$, $(0, -b)$, where $a$ and $b$ are positive ...
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1answer
21 views

Rotating point by angle

Let $X = (c, 0)$. If I will rotate $X$ by, say, angle $\alpha = \frac{\pi}{4}$, how can I determine position of new angle? Will it just be $X' = (c + \cos\alpha, \sin\alpha)$?
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1answer
12 views

Degree measure of multiple polygons

I made this design on the Desmos calculator, and I was wondering what the quickest way was to find the degree measure of each individual angle. What I know so far: The measures of each of the ...
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3answers
101 views

Area of extended triangle [closed]

I have three points $$(0,0),\ (1,1),\ (2,0)$$ and $k$, where $k$ is a number, in this task $k = 2$. I need to calculate the area of ​​the figure extending it points less than or equal to $k$. (In ...
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1answer
52 views

Calculate area of a figure extended from the unit square

I have four points $$(0,0),\ (0,1),\ (1,1),\ (1,0)$$ and $k$, where $k$ is a number, in this task $k = 1$. I need to calculate the area of ​​the figure extending it points less than or equal to $k$....
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0answers
25 views

Learn tracing of conics and concoids

A major portion of my course revolves around tracing of conics and concoids. But the explanation in my books is poor. I'm looking for some online notes/texts or videos to learn tracing of curves. I ...
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1answer
29 views

Tips to fix coordinates in analytic geometry.

I now know how useful analytic geometry can be in bashing geometry problems involving side lengths. Does anybody have any tips on how to fix coordinates to keep the solution from becoming too tedious? ...
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1answer
29 views

Graphing a regular pentagon

I just realized that I didn't know how to graph a regular pentagon with integer coordinates... What are some possible coordinates for a regular pentagon with the uppermost point at coordinate (0,0)?
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0answers
16 views

eccentricity of the conic

I'm given this question to find the eccentricity of this conic : $x^2 + ky = 0, k>0$ The given equation can be written as $x^2 = -ky$ now we can say compare this with the equation of parabola. But ...
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0answers
12 views

triangulation of a surface, adapted to curvature

This is about my printed models of mathematical objects. All of the designs that I've published so far consist of grids of bent ‘rods’, and in most of them the spacing of vertices depends on the rod'...
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2answers
24 views

A $2\times 2$ linear matrix transformation is conformal if and only if $c=-b ,d=a$ and $a,b \neq 0$

Let $T:\mathbb{R}^{2} \to \mathbb{R}^{2}$ a linear transformation represented by the matrix $A=\begin{pmatrix} a & c \\b&d \end{pmatrix}$ . Show that $T$ is conformal if and only if $c=-b ,d=a$...
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1answer
49 views

Equilateral triangle with vertices whose coordinates on the Cartesian plane are integers. Does such a triangle exist? [duplicate]

Can you build an equilateral triangle on a Cartesian plane whose vertices only have integer values as their coordinates? Looking at the simplest example, i.e. a triangle with vertices (0,0), (1,0) ...
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2answers
86 views

Prove that the value of $(abc)-(ab+bc+ca)+3(a+b+c)$ is $0$

If the points $\big(\frac{a^3}{a-1}, \frac{a^2-3}{a-1}),(\frac{b^3}{b-1}, \frac{b^2-3}{b-1}) ,\big(\frac{c^3}{c-1}, \frac{c^2-3}{c-1}\big)$ are collinear for three distinct values of $a,b,c$ and $a,b,...
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1answer
32 views

parameterization of polar coordinates vectors

Assuming everything in 2D, if there is a circle with centre at origin, with radius R , we can write its cartesian equation as : x^2 + y^2 = R^2 . It's vector form in cartesian coordinates will be : $$...
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0answers
20 views

In how many ways does a vector determina a hyperplane?

The perpendicular hyperplane is the *standard* way. The gradient of a differentiable function $f: \Bbb R^{n+1} \rightarrow \Bbb R$ gives another idea. Select one direction to be the height $h$. Then, ...
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3answers
37 views

If the reflection of the hyperbola $xy = 4$ in the line $x - y + 1 = 0$ is $xy = mx + ny + l$ find $m + n + l$

If the reflection of the hyperbola $xy = 4$ in the line $x - y + 1 = 0$ is $xy = mx + ny + l$ find $m + n + l$. I already solved it by taking a general point (more than one way possible for it) and ...
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1answer
35 views

Point P such that perimeter is least

Given two points $A(-2,0)$ and $B(0,4)$ then find coordinate of point $P$ lying on the line $2x-3y=9$ so that perimeter of triangle $APB$ is least. Doing it by traditional calculus is making ...
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1answer
40 views

How to calculate triangle coordinates in cartesian plane?

My problem can be describe by following image: I know coordinates of an example P point. Say, they are equal to (8,8). I also ...
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0answers
25 views

prove that If and only if lines are perpendicular, the slopes are negative reciprocal.

I have to prove that If and only if lines are perpendicular, the slopes are negative reciprocal. I know to prove that if the lines are perpendicular,the slopes are negative reciprocal. But I dont ...
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3answers
40 views

If a family of straight lines is $\lambda^2 P+\lambda Q+R=0$ ,then the family of lines will be tangent to the curve $Q^2=4PR.$

I have read this theorem in my book but i do not know how to prove it. If a family of straight lines can be represented by an equation $\lambda^2 P+\lambda Q+R=0$ where $\lambda $ is a parameter and $...
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1answer
33 views

Focus of the Parabola

Find the Focus of $$(2x+y-1)^2=5(x-2y-3)$$. Clearly its a Parabola whose axis is $2x+y-1=0$ and since $x-2y-3=0$ is perpendicular to $2x+y-1=0$ Tangent at the vertex is $x-2y-3=0$.Also the Vertex is $...
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1answer
17 views

Would $f(x,y) = \frac{y}{x}$ give an actual graph of all the possible slopes of a function of 1 variable?

If you let $y$ stand for $\Delta y$ and $x$ stand for $\Delta x$. I wanted to plot this because I thought it would be interesting among other things to see the behavior of the function around $x=0$. ...
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2answers
39 views

what is the shortest distance between a parabola and the circle?

what is the shortest distance between the parabola and the circle? the equation of parabola is $$y^2=4ax$$ and the equation of circle is $$x^2+y^2-24y+81=0$$ if you can show graphically it will be ...
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1answer
125 views

Shortest distance between two circles

What is the shortest distance, in units, between the circles $(x - 9)^2 + (y - 5)^2 = 6.25$ and $(x + 6)^2 + (y + 3)^2 = 49$? Express your answer as a decimal to the nearest tenth. So I know that ...
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2answers
46 views

the equation of two sides of a parallelogram are $2x-3y+7=0$ and $4x+y-21=0$ and one vertex is $(-1,-3)$. Find the other vertices.

First, I checked if the point $(-1,-3)$ is not a solution to the two given equations above so therefore none of those lines passes that point. Then, I solved for the lines parallel to equations above ...
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1answer
26 views

Line $mx + ny = 3$ is normal to the hyperbola $x^2 – y^2 = 1$

If the line $mx + ny = 3$ is normal to the hyperbola $x^2 – y^2 = 1$, then evaluate $\frac{1}{m^2}+\frac{1}{n^2}$. I compared given equation of normal to equation of normal at parametric point i.e $x\...
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2answers
26 views

The number of circles passing through the vertices of a triangle

I have read a book written by C.V Durell on Geometry. In this book I have found a lemma which states that there is one and only one circle that passes through three vertices of a triangle. I thought ...
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1answer
27 views

Draw a plane through a line parallel to the $x$-axis

Can someone help me with this problem? I bet it will be pretty easy for the most of you: Through the line $p$ draw a plane that is parallel to the $x$-axis, where p is defined by: $x=5-2t, y=9t-...
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1answer
22 views

Equation of plane passing through intersection of line and plane

Find the equation of the plane passing through the intersection of line $$\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{2}$$ and the plane $$x-y+z=5$$ and parallel to a vector with direction ratios $<2,3,...
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3answers
30 views

Show that the equation of the tangent to the parabola $y^2=4ax$ at the point (p,q) is $qy=2a(x+p)$

Question: Show that the equation of the tangent to the parabola $y^2=4ax$ at the point (p,q) is $qy=2a(x+p)$ These are my two approaches: First approach: If we have $(p,q)$ as $(x_1,y_1)$ $$y^...
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1answer
22 views

Equation of an $(n-2)-$sphere in $\mathbb{R}^n$.

I am looking for the equation of an $(n-2)$-sphere in $\mathbb{R}^{n}$ generated from the intersection of the $(n-1)$- sphere $x_1^2 + x_2^2 + \cdots + x_n^2 = r^2$, and the hyperplane perpendicular ...
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1answer
26 views

Is there any book online that shows mathematical procedures relating to perspective drawing?

I am trying to learn comprehensive mathematical analyses (rather than geometrical methods) about perspective drawing projections. Can anyone suggest a good online book to buy that illustrates ...
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2answers
84 views

Find area bounded by parabola $y^2=2px,p\in\mathbb R$ and normal to parabola that closes an angle $\alpha=\frac{3\pi}{4}$ with the positive $Ox$ axis.

Let $p=-2<0\Rightarrow y=\sqrt{-4x} \lor y=-\sqrt{-4x}\Rightarrow x\le 0 $. Let $p=2>0\Rightarrow y=\sqrt{4x} \lor y=-\sqrt{4x}\Rightarrow x\ge 0 $. For $p>0$ we can find the equation for ...
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3answers
49 views

Equations of the line which intersects the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x+2}{1}=\frac{y-3}{2}=\frac{z+1}{4}$

Find the equations of the line which intersects the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x+2}{1}=\frac{y-3}{2}=\frac{z+1}{4}$ and passes through the point $(1,1,1)$. First I ...
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0answers
25 views

Find functions $\xi_1(x)$ and $\xi_2(x)$ and scalars $\alpha, \beta \in \mathbb{R}$ to characterize a set

Find functions $\xi_1(x)$ and $\xi_2(x)$ and scalars $\alpha, \beta \in \mathbb{R}$, such that $\xi_1(x) \le y \le \xi_2(x)$ when $\alpha \le x\le \beta$ equivalent to to the following set in $\...
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1answer
31 views

lattice point on a circle

consider a circle with center (sqrt[2],1/3) and any arbitrary radius. how do I prove that there is atmost one lattice point on the circle? also, does there exist an unique cirle with exactly 2004 ...
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0answers
14 views

Geometric interpretation of zeros of differential of complex polynomial

I want to show that if three complex numbers $ a,b,c$ don't lie on the same line, then if $p,q$ are such that $W^{'}(p)=0=W^{'}(q)$ where $W(z)=(z-a)(z-b)(z-c)$ then angles $acp,bcq$ are equal.I have ...
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3answers
75 views

Curve-fitting using circles

I'm working for a firm, who can only use straight lines and (parts of) circles. Now I would like to do the following: imagine a square of size $5\times5$. I would like to expand it with $2$ in the $x$...
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2answers
24 views

Determine whether two segments P1Q1 and P2Q2 have a common point if the (x,y) coordinates of their end points is known?

Does this question have a solution? I think it's impossible to know if line segments P1Q1 and P2Q2 intersect at all with just the information about their end points Q1 and Q2. Thanks.
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3answers
75 views

Reflection of a Light Ray

I found this problem to be very hard while studying for the exam: Let $$L: \vec r(t)=<1,-2,3>+t<-5,4,1>, \qquad t \in \mathbb{R}$$ be a line. Light is traveling along the line $L$ ...
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1answer
53 views

Coordinate Geometry finding x and y

How would I rearrange this equation to find $x_3$ and $y_3$ $$\tan\ \alpha =\frac{\sqrt{(x_3-x_2)^2+(y_3-y_2)^2}}{ \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}}$$ EDIT: So basically what I want to do is that I ...
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1answer
18 views

Limiting points subtend right angle at the centre

If the limiting points of the system of circles $x^2+ y^2+ 2gx +w(x^2+ y^2+ 2fy + k)=0$ where $w$ is a parameter , subtends a right angle at origin then find value $k/f^2$? I know that limiting point ...
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1answer
53 views

Three circles intersect at one point.

If three circles intersect at one point then there's unique $x$ and $y$ coordinate values such that the following equations are satisfied: $$(x-x_i)^2 + (y-y_i)^2 = r_i^2$$ Where $i=1,2,3$ Taking ...
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1answer
43 views

Relation between $a$ and $b$ when equation of obtuse angle bisector is $ax+by-3=0$

The combined equation of bisector of angles between the lines $L_1$ and $L_2$ is $$2x^2-3xy-2y^2-x+7y-3=0$$ $P(4,-3)$ is a point on $L_1$. If the equation of obtuse angle bisector is $ax+...
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1answer
38 views

Systems of Equations (Inconsistent)

Question: Consider the following system of three equations: $$2y+2z=9-2x$$ $$x=12-3y-4z$$ $$Ax+5y+6z=B$$ Find values of A and B which makes the system of equations ...