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

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Alternate form for the vector field $x^3\cdot\hat{x} + y^3\cdot\hat{y}$

Is there an alternate form for the vector field $x^3\cdot\hat{x} + y^3\cdot\hat{y}$ in which we can write all in function only of the radius $r=\sqrt{x^2 + y^2}$ ? Thank you
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2answers
30 views

Given $\Vert \vec{u} \Vert$ and $\Vert \vec{v} \Vert$ and $\angle 120^\circ$ find volume with sides $\vec{u} \times \vec{v}$, $\vec{u}$ and $\vec{v}$

I am given the following problem: Knowing that $\Vert \vec{u} \Vert = 3$ and $\Vert \vec{v} \Vert = 4$ and also $\angle (\vec{u}, \vec{v}) = 120^\circ$ find the volume of the parallelepiped with ...
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2answers
69 views

If $x^2=\lambda$, then find the value of $\lambda$

A Circle $C_1$ is drawn having any point $P$ on $X$- axis as its centre and passing through the centre of the circle $C: x^2+y^2=1$. A common tangent to $C_1$, and $C$ intersects the circle at ...
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1answer
34 views

Finding straight lines

I am new to analytical geometry and excuse me for my notations. We have four lines: $l_1: u_1x + v_1y + r_1 = 0$ $l_2: u_2x + v_2y + r_2 = 0$ $m_1: u_3x + v_3y + r_3 = 0$ $m_2: u_4x + v_4y + r_4 = ...
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1answer
29 views

Find the centre of a circle circumscribing the triangle whose angular points are $(1,1), (2,3), (-2,2)$

The main question is as follows : Find the point $P$, such that $P$ is the centre of a circle circumscribing the triangle whose angular points are $(1,1), (2,3), (-2,2)$. My method : I am ...
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73 views

Great mathematical fusions in math history

Development of the mathematics resembles usually a growing tree - from old branches grow new ones. However sometimes domains of mathematics which were separated for the long time are fused together ...
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1answer
18 views

If tangents are drawn from two points which are equidistant from given point, then find the locus

Tangents are drawn to the circle $x^2+y^2=a^2$ from two points on the $X$ axis equidistant from the point $(k,0)$ prove locus of their intersection is $ky​^2=a^2(k-x)$. If I take points as $(k+\alpha,...
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1answer
34 views

How to prove the parallel projection of an ellipsoid is an ellipse?

Take the following ellipsoid in implicit form as an example: $$x^2 + 2 y^2 + 3 z^2 + x y + y z - 2 xz = 5$$ which shows: The parallel projection of the ellipsoid onto $xoy$ coordinate plane can ...
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1answer
90 views

Is the smallest ellipsoid enclosing a convex set unique?

Let $S \subset \mathbb{R}^n$ be a convex set. Assume that it is bounded. We want to find an ellipsoid $E$ of smallest volume such that $S \subset E$. Is $E$ unique?
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20 views

Standard form of an Ellipse Given…

So I'm currently stuck on how to get the standard form of the equation of the ellipse given the characteristics Vertical Major Axis and passes through the points ( 0,6 ) and ( 3,0 ) Any Ideas? ...
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1answer
36 views

Radius of the sphere inscribed at the corner of an irregular polyhedron

Does anyone know the mathematical formula to calculate the radius of a sphere that can be inscribed at the corner of an irregular polyhedron ? There can be several radius, but the sphere should not ...
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0answers
59 views
+50

'Tetrahedral' coordinates in space (generalization of hexagonal coordinates)

The Cartesian coordinates are the most widely used in Euclidean space of any dimension. However, there is another set of coordinate systems which can in some way be considered optimal. Imagine ...
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1answer
48 views

Circle Problem:Which of the following are true

If the circle $x^2+y^2+2gx+2fy+c=0$ cuts the three circles $x^2+y^2−5=0$, $x^2+y^2−8x−6y+10=0$ and $x^2+y^2−4x+2y−2=0$ at the extremities of their diameters, then which of the following are true ? $c=...
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11 views

Generating an Equation for Quadratic Bezier Curve

So I know that a standard equation for a quadratic Bezier Curve is: $C(t)=(1-t)P_0+2t(1-t)P_1+t^2P_2$ I have been asked to generate an equation to model my quadratic bezier curve. I known that my $...
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1answer
25 views

Find the coordinates of P and Q.

A line is drawn through the point $A(1,2)$ to cut the line $2y=3x-5$ in $P$ and the line $x+y=12$ in $Q$. If $AQ=2AP$, find the coordinates of $P$ and $Q$. From: Mathematics, The Core Course for A-...
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1answer
26 views

Is there a simple way to decide if a hyperboloid is one-sheeted or two-sheeted, given the quadric equation?

Let us say that we have a quadric equation, whose solution set lies in $\mathbb{R}^3$, and you know it's a hyperboloid. Is there a way to analytically decide through a criterion if the hyperboloid is ...
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22 views

Clarification Needed:Equation Of Refracted Ray/Line

A ray of light is sent along the line $2x-3y=5$.After refracting across the line $x+y=1$ it enters the opposite side after turning by $15^0$ away from the line $x+y=1$.Find the equation of the ...
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2answers
9 views

Given $\angle (\vec{u} , \vec{v}) = 30^\circ$, $\Vert \vec{w} \Vert = 4$ and $\vec{w}$ is $\perp$ to both find $\vec{u} \cdot \vec{v} \times \vec{w}$

I am given the following problem: Knowing that the angle between the unit vectors $\angle (\vec{u} , \vec{v}) = 30^\circ$, $\Vert \vec{w} \Vert = 4$ and that $\vec{w}$ is orthogonal to both of ...
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1answer
14 views

Given the tethraedron $OABC$ find ratio between its Volume and $\vert \vec{OM} \cdot \vec{ON} \times \vec{OP} \vert$

Given the tethraedron $OABC$ where $\vec{OA} = \vec{a}$, $\vec{OB} = \vec{b}$ and $\vec{OC} = \vec{c}$ and the points $M$, $N$ and $P$, which are the midpoints of segments $\vec{AC}$, $\vec{AB}$ and $\...
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Sum of slopes between points of concurrent normals to hyperbola is zero

Let $ A_1, A_2, A_3, A_4 $ be four points on the hyperbola $xy = 1$. Suppose that the normals to the hyperbola at these four points are concurrent, i.e. they intersect in a single point. Prove that ...
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1answer
34 views

Given that $ \{ \vec{u}, \vec{v} \}$ are l.i. prove that if $ \vec{w} \times \vec{u} = \vec{w} \times \vec{v} = \vec{0}$ then $\vec{w} = \vec{0}$

I am asked to elaborate on the following proof: Given that $ \{ \vec{u}, \vec{v} \}$ are linearly independent prove that if $ \vec{w} \times \vec{u} = \vec{w} \times \vec{v} = \vec{0}$ then $\vec{...
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2answers
50 views

Using cross product prove that if $\vec{u} \times \vec{v} = \vec{0}$ and $\vec{u} \cdot \vec{v} = 0$ then $\vec{v} = \vec{0}$

I am asked to elaborate on the following proof: Let $\vec{u} \neq \vec{0}$. Prove that if $\vec{u} \times \vec{v} = \vec{0}$ and $\vec{u} \cdot \vec{v} = 0$ then $\vec{v} = \vec{0}$. My attempt ...
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6answers
80 views

Checking nature of angles of a triangle given the equations of the three lines that form a triangle

Suppose we have three lines $\ell_i=a_ix+b_iy=c_i$, $i=1,2,3$ and we are given that they form a triangle. I need to find which angles are acute and which are obtuse without plotting the lines ...
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1answer
1k views

IMO 2016 Problem 3

Let $P = A_1 A_2 \cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \ldots, A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $...
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1answer
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Pair of straight lines problem: Prove that $g (a_1+b_1)=g_1 (a+b) $

If the lines joining the origin and the point of intersection of the curves $ax^2+2hxy+by^2+2gx=0$ and $a_1x^2+2h_1xy+b_1y^2+2g_1x=0$ are mutually perpendicular then prove that $g (a_1+b_1)=g_1 (a+b) $...
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1answer
41 views

Prove that locus of orthocentre is $x+y=0$

The base of a triangle is axis of $x$ and its other $2$ sides are given by the equations :$y = (1+α)\frac{x}{α} + (1+α)$ and $y = (1+β) \frac{x}{β} + (1+β)$.Prove that the locus of its orthocentre is ...
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1answer
20 views

Show that, if $u,v,w$ are orthogonal two-by-two, then $S = \{ u , v , w\}$ forms a basis which is linearly independent

I am given the following question: Show that, if $u,v,w$ are orthogonal two-by-two, then $S = \{ u , v , w\}$ forms a basis which is linearly independent. My idea to tackle this problem is to ...
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3answers
39 views

Verify if $\overrightarrow{w}$ is a linear combination of $\overrightarrow{u}$ and $\overrightarrow{v}$

I am given the following question: Let $\Vert \overrightarrow{u} \Vert = \Vert \overrightarrow{v} \Vert = \Vert \overrightarrow{w} \Vert = 1$ and $\overrightarrow{u} \cdot \overrightarrow{v} = \...
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1answer
19 views

intersection of a line (certain direction) and a circle

I need to calculate (previously) the point where a ball will touch the inside of a circle (for a game I'm developing). So I have two equations, one of the direction of the ball, and another of the ...
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7answers
551 views

Get the equation of a circle when given 30 points [closed]

A similar question has been asked before on this site but that was of getting equation of circle using 3 points. I want my center to be more accurate So my question is how can i get the center of ...
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1answer
36 views

Dimensions of bounding box for arbitrary circle sector

I need to determine the dimensions of bounding box for arbitrary circle sector as shown in the diagram below. Given: φ = Start angle in the range of 0 ~ 2π θ = Sweep angle in the range of 0 ~ 2π r =...
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1answer
26 views

Analytical geometry - Finding the coordinates of point M

I've been practicing analytical geometry lately and I've come to a problem. I solved the problem a few times but I can't get the right result. Here is the math problem: Point M whose distance ...
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2answers
43 views

Largest bifocal triangle in an ellipse

Ellipse $E$ has foci at $P$ and $Q$, and semi-major and semi-minor axes of length $a$ and $b$, respectively. Find the area of the largest triangle that can be (parttially) inscribed in ellipse $E$, ...
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2answers
41 views

Volume calculation with change of variables

I am trying to calculate the volume of a solid, given the equations of its bounding surfaces. It is a $3$-dimensional object, so the equations are in $x$, $y$ and $z$. In order to simplify the ...
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1answer
18 views

Calculate equally-spaced and arranged center points on the edges of a circle based on its diameter and grid origin?

Question for a project: I have a circle that I know the diameter of (and therefore also the height/width dimensions of on a grid based on its top/left origin X/Y)... but I wanted to calculate equally ...
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1answer
37 views

graph of $z=2x+y$

The usual technique using traces where one variable is set to 0 does not seem to work here since I get all 0's and so where do the intersections meet? I looked in my calc. book and the technique ...
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4answers
99 views

Graph of the function $\cos(x)\cos(x+2)-\cos^2(x+1)$ will be?

Graph of the function $\cos(x)\cos(x+2)-\cos^2(x+1)$ will be? (A)A straight line (B)A parabola Give the corresponding equation too. Source:JEE 1997. Can ...
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0answers
38 views

Equations of the tangent planes to the sphere

Find the equations of the tangent planes to the sphere $x^2+y^2+z^2-10x+2y+26z-113=0$ which are parallel to the straight lines $\frac{x+5}{2}=\frac{y-1}{-3}=\frac{z+13}{2}$ and $\frac{x+7}{3}=\frac{y+...
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1answer
17 views

2D AABB count vector for not having collision after movement

I hope it is correct here, I feel like this question is more math related than programming. Table of Contents Introduction What is question about / problem description What way I figured out Other ...
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0answers
13 views

Condition for n points in the plane to determine a convex n gon

Suppose there are n points in the plane, labelled 1 through n, no three of which lie on a line. Suppose further that for every triple [i,j,k] with i< j < k that travelling from i to j to k is ...
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2answers
44 views

Questions regarding dot product (possible textbook mistake)

I am given the following exercise: Show that $\Vert \overrightarrow{a} + \overrightarrow{b} \Vert = \Vert \overrightarrow{a} \Vert + \Vert \overrightarrow{b} \Vert $ if and only if $\...
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1answer
44 views

Necessary & Sufficient condition for the line $ax+by+c=0$ to pass through the 1st quadrant

What is the necessary and sufficient condition for the line $ax+by+c=0$, where $a,b,c$ are non-zero real numbers, to pass through the first quadrant? I could find the points at which the line ...
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44 views

The locus of the orthocentre of the triangle formed by the lines

The following question is from IITJEE 2009 paper. The locus of the orthocentre of the triangle formed by the lines $(1 + p)x − py + p(1 + p) = 0$ $(1 + q)x − qy + q(q + 1) = 0$ and $y = 0$, where $...
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0answers
15 views

Pinhole projection of the center of a 3D circle

Consider the pinhole projection of a 3D circle. The projection I am considering is a pinhole camera projection which has a fully known calibration. The projection of a 3D circle will be an ellipse, ...
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1answer
22 views

To find the centre of the inner circle that is tangent to the unit circle and the x-axis

We have a unit circle $C:x^2+y^2=1$. Let $l:y=m(x+1)$. We consider a circle $C'$ at a centre on $l$ that is inscribed to an upper semi-circle, i.e., a circle that is tangent to the circle $C$ and the ...
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4answers
35 views

Proving that the lines cut the coordinate axes in concylic points

The lines $2x +3y +19 = 0$ and $9x+6y-17 = 0$ cut the coordinate axes in concyclic points.What would be the fastest method to prove it manually?Is it possible to prove the statement without having to ...
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0answers
11 views

Split a map into roughly equal sections directionally and put points in it

I have a 16000 x 9000 grid map and I want to split it into x sections that are preferably of equal size. Then I want to place points on each section are centers of circles with a 2200 unit radius and ...
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0answers
76 views

Number of $N$ formed from the set of points

Given $k$ points on 2d plane, I need to find the number of $N$ shaped figures from these $k$ points. lets consider four different points from the set and name them $A$, $B$, $C$, and $D$ (in that ...
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1answer
62 views

Ellipse and parallel lines

Let's imagine that we have an ellipse described by the known equation $v^TAv=0$, (Link_1) where $v=[x \ y \ 1]^T$ (it can be a skew one in a general case). Then we have all possible parallel lines - ...
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1answer
10 views

Find point where a line of multiple vertices overlaps itself

Since I'm not familiar with a lot of mathematical terminology, I will explain this problem with a little story. Imagine you and your friend Anne have a piece of string each, and place it on a ...