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

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Lines intersections distance on the asymptotes

Like in picture we have two lines. Lenght of one of them is 2E and other's lenght 2C and also ellipse asymptotes are A and B and its center is on origin(0,0) I want to find D and F How can I ...
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1answer
25 views

A specific case of quadratic forms

I have a quadric as follows: $$ax^2+by^2+bz^2+yz=0.$$ I am curious to know which shapes in $\mathbb{R}^3$ this equation describes for different value of $a$ and $b$?
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Angle of the tangent vector of a parabola in function of the angle of the vector that defines this parabola (Apostol, chapter 14.21, problem 1)

Apostol, chapter 14.21, problem 1 (a review problem) Here is the question: Let r denote the vector from the origin to an arbitrary point on the parabola $y^2 = x$, let $\alpha$ be the angle that ...
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0answers
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4 points in 3-d space (one known and three unknown)

Problem in 3-d space. We have four points: $P_0$ where we know coordinates $(0,0,0)$ and $P_1, P_2, P_3$ where coordinates are unknown. However we know distances between $P_1, P_2, P_3$ (let's name ...
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1answer
14 views

How to points of line in ellipse when it's moved as ellipse tangent [on hold]

Ellipse Picture In the picture minor and minor asymptote and points of line (X&Y) are known, when we move the line new position is X' and Y'. How can be calculated new position of line or is ...
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0answers
11 views

Same Center Ellipse Major and Minor Axes

Ellipse Picture I have two same center ellipses A, B, and C are known values X and Y arent known values and I need to obtain these values. How can it be calculated?
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2answers
23 views

Calculate the plane equation of 2 vectors. [on hold]

Which type should I use in order to calculate the plane equation that is defined by 2 vectors, let's say V1 $\langle{1,2,3}\rangle$ V2 $\langle{4,5,6}\rangle$.
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24 views

How to find the Coordinate equation of a curve which bends all the parallel rays from infinity towards a single point

How should I proceed on to find the coordinate equation of a curve such that it bends all the parallel rays coming from infinity towards a single point. Yes I know that it would be a 2nd degree ...
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11 views

Мöbius Transformations and circle inversion

Can a Möbius Transformation be decomposed into a composition of 2 generalized circle inversions?
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1answer
19 views

Proof of the reflective property of the ellipese

I'm trying to prove the reflection property of the ellipses for an optics problem. The property is that that a ray of light originated at one of the ellipse's foci reflects in such a way to pass ...
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4answers
46 views

Is this a correct way to solve this high school coordinate geometry question?

Here's the question: Given point $A$: $(-3;-1)$ Given point $B$: $(3;7)$ Given point $Z$: $(x;0)$ Find the $x$ coordinate of point $Z$ so that the angle of view of AB segment is $90$ ...
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1answer
14 views

Finding 3rd circle's coordinate of particular radius given 2 circles coordinate, circles touch externally

Given circle say A,B,C where each of them touches each other externally . We are given radius of all 3 circles. We are also given 2-D coordinates of centre of B,C ,we need to compute coordinates of A. ...
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2answers
62 views

Different methods for finding the minimum of $|x-2y|$ when $x^2+1=2y^2$.

For $x, y \in \Bbb R$, $x^2 + 1 = 2y^2$, find the minimum of $|x - 2y|$. At a glance I found that the point $(x, y)$ lies on a hyperbola and $|x - 2y|$ is just the distance between the point and the ...
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1answer
32 views

Getting the coordinates of the center of a circle bisecting two other circles.

We have circles $C_1$ and $C_2$ with centers $(-d,0)$ and $(d,0)$, radii $a_1<d$ and $a_2<d$ respectively. If circle $D$ with radius $r$ (and with centre not necessarily on the x-axis) bisects ...
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0answers
14 views

Linear functionals and hyperplanes

If $L:\Bbb R^n\to\Bbb R$ is a non-trivial linear functional , i.e $L(x+y)=L(x)+L(y), x,y \in\Bbb R^n$ and $L(ax)=aL(x), x \in\Bbb R^n, a \in\Bbb R$, then why does the set of all x $\in\Bbb R^n$ that ...
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37 views

Finding $x^2$ and $y^2$ of hyperbola

Currently, I am trying to the $x^2$ and $y^2$ of a hyperbola. I have the vertices at $(-1, -1)$ $(5, -1)$ I have the focus at $(-4, -1)$ $(8, -1)$ I know that the distance between two vertices ...
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1answer
34 views

the area that a part of an ellipse consumes in a square of a discrete grid

Think about a discrete grid of unit 1, which means the grid consists of infinite number of squares whose area is 1. You can assign a coordinate to each square and one of them will have the coordinate ...
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42 views

Given any parametric curve, finding its general form?

I'll illustrate the problem I'm trying to solve with an example. Let's consider the equations $$ x = \cos (t) $$ $$ y = \sin (t) $$ We know that these are a parametric form of the unit circle. In ...
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1answer
34 views

Formula for area of triangle in complex plane [closed]

If $A(z_1)$, $B(z_2)$, $C(z_3)$ are vertices of a triangle $ABC$ in Argand plane, what is the area of the triangle?
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1answer
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Plot points on an arc

I have modified this post with updated information so the problem may be more clear. Because the answer provided does not achieve the results intended, maybe adjusting the content will help adjust ...
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1answer
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pls help to the sum [closed]

A plane passes through a fixed point $(a,b,c)$. Show that the locus of the foot of perpendicular to it from origin is the sphere $x^2+y^2+z^2-ax-by-cz=0$
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1answer
27 views

Find the point on the plane xOy [closed]

Let $A(x_1; y_1)$, $B(x_2, y_2)$ and $C(x_3, y_3)$ be three points not lying on the same straight line. Find the point on the plane $xOy$ such that the sum of the distances from it to these points is ...
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22 views

Find a linear sequence of the common terms between 2 other sequences?

Given 2 linear sequences $an_1+b$ and $cn_2+d$ generate a sequence of whole numbers that can expressed as $an_1+b$ and $cn_2 + d$. An example to illustrate this: Given $2n+3$ and $3n+6$ the sequence ...
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2answers
36 views

Proof that if two lines are parallel then $A_1$ = $A_2$ and $B_1$ = $B_2$?

Let two lines to be parallel in their general form. $L_1$ : $A_1 x$ + $B_1 y$ + $C_1$ $L_2$ : $A_2 x$ + $B_2 y$ + $C_2$ Now i wish to prove $A_1$ = $A_2$ and $B_1$ = $B_2$ But i can only think of ...
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2answers
29 views

Slope of axes of a General Conic Section

A General Conic Section is given by the equation $ax^2 + by^2 + 2hxy +2gx +2fy + c =0 $. Let the $\theta$ be the slope of one of its axes. Prove that : $$\tan 2\theta = ...
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1answer
21 views

Hyperbolas and Quadrants on Rotation

Let's assume we have a standard hyperbola. On rotating the hyperbola $45^{\circ}$ clockwise, the new hyperbola should lie in the $2$nd and $4$th quadrant. However, the equation of a parabola rotated ...
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1answer
31 views

Coordinates of incentre without finding side lengths

If I am given the equations of sides of a triangle and I need to find incentre what is the shortest method ? Is it possible without having to find lengths of sides of triangle?
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1answer
32 views

Vectors: Using Pythagoras's theorem for magnitude in the 4th dimension

For a simple x and y plane (2 dimensional), to find the distance between two points we would use the formula $$ a^2 +b^2 = c^2 $$ For a slightly more complicated plane; x,y and z (3 dimensional), ...
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2answers
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Suppose you graphed every single point of the form (2t + 3, 3-3t).

Suppose you graphed every single point of the form $(2t + 3, 3-3t)$. For example, when $t=2$, we have $2t + 3 = 7$ and $3-3t = -3$, so $(7,-3)$ is on the graph. Explain why the graph is a line, and ...
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1answer
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Points on parabola with abscissa in A.P. and ordinate in G.P.

The points with coordinates $(a,b),(a_1,b_1),(a_2,b_2)$ are points on parabola $y=3x^2$. The numbers $a,a_1,a_2$ are in Arithmetic progression while $b,b_1,b_2$ are in Geometric Progression. Calculate ...
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1answer
18 views

General conic equation and coefficient matrices

For a general conic $Q(x,y)=ax^2+2hxy+by^2+2gx+2fy+c$ we define a matrix $A$ as follows: $A=\left( \begin{matrix} a& h& g\\ h& b& f\\ g& f& c\end{matrix} \right)$. Then we ...
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1answer
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Show that the three distinct points $(p,p^2)$, $(q,q^2)$ and $(r,r^2)$ can never be collinear.

Show that the three distinct points $(p,p^2)$, $(q,q^2)$ and $(r,r^2)$ can never be collinear. I can think of the graph $y=x^2$ to solve the above problem graphically. However, I wanted to solve it ...
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A pair of tangents to a conic intercepts 2k on y axis. Find locus of their point of intersection.

A pair of tangents to the conic $ax^2 +by^2 = 1$ intercepts a constant distance 2k on the y axis. Prove that the locus of their point of intersection is the conic: $$ax^2(ax^2 + by^2 -1) = bk^2(ax^2 - ...
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283 views

Geometric interpretation for eigenvalues and eigenvectors of the cross product's representation as a linear map

Fix ${\bf x} = (x_1,x_2,x_3) \in \Bbb R^3\setminus\{{\bf 0}\}$. We can look at the cross product as a linear map ${\bf x}\times: \Bbb R^3 \to \Bbb R^3$ which is represented in the standard basis by ...
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Find the equation of a hyperbola, given a point on it and the length of the transverse axis

My textbook has the following question: The transverse axis of a hyperbola is of length $24$ and the curve passes through the point $(13, 10)$. Find the equation of the hyperbola. Also give the ...
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1answer
34 views

How to derive formula for focus of a parabola?

I understand how to obtain the formula for the vertex of a formula, $ y= a(x-h) + k $ where $ h=-b/2a$ and the vertex is $(h,k)$. However I don't know how to get to $(h,k+1/4a)$. Could someone please ...
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2answers
55 views

Ellipse - relation between a and b such that $F_1P \perp F_2P$

Consider the ellipse $\displaystyle \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}=1$ with foci $F_1 (-e, 0)$ and $F_2 (e, 0)$ (where $e$ is the linear eccentricity). What is the relation between $a$ and $b$ so ...
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24 views

Homogenization of Equations

Say there are two equations: $3x^2+mxy-4x+1=0$ and $2x+y-1=0$. I have to find possible values of $m$ for which lines joining the points of intersection of above two equations are at right angles. I ...
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Spherical coordinates after rotations in 3D

It's straightforward to derive rotation matrices in 3D space around the x, y and z axes. Those matrices give the new coordinates x', y' and z' in terms of the old components x, y and z and the angle ...
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Can $n$ circles be drawn such that all have a common intersection but no two intersect individually

I was fiddling with plane geometry when a question came into my mind: Can $n$ circles ($n \ge 3$, $n \in \mathbb{N}$) be drawn such that: $C_1 \cap C_2 \cap C_3 \cap \ldots \cap C_n \not = ...
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Vector Distance

let there be a line L: $\frac{x-1}{2}= \frac{y+1}{3}= \frac{z}{1}$ and a plane: $2x-y-z=5$. With this given data find: a line L1, such that L1 is parallel to L, is in P, and the distance between L and ...
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$(x-1)(y-2)=5$ and $(x-1)^2+(y+2)^2=r^2$ intersect at four points $A,B,C,D$. Centroid of $\Delta ABC$ lies on $y=3x-4$, then the locus of $D$

$(x-1)(y-2)=5$ and $(x-1)^2+(y+2)^2=r^2$ intersect at four points $A,B,C,D$. If centroid of $\Delta ABC$ lies on $y=3x-4$, then what is the locus of $D$? I did try a couple of things, but I honestly ...
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How to determine if a point lies in this particular convex region?

I have a family of hyperplanes which do not contain the origin: \begin{eqnarray} a_{11}x_1+a_{12}x_2+\dots+a_{1n}x_n &=& k_1\\ a_{21}x_1+a_{22}x_2+\dots+a_{2n}x_n &=& k_2\\ ...
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4answers
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Check if a given coordinate lies in path of a ray (coordinate geometry)

As shown in the image I have two known coordinate pair A and B and few other known coordinate pairs (RED blob) on the graph. I need to know if any of the other given coordinates fall in line of the ...
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Cosine Inequality

Show that given three angles $A,B,C\ge0$ with $A+B+C=2\pi$ and any positive numbers $a,b,c$ we have $$bc\cos A + ca \cos B + ab \cos C \ge -\frac {a^2+b^2+c^2}{2}$$ This problem was given in the ...
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1answer
36 views

Where i am going wrong in finding normal to curve?

The question is Find the perpendicular distance between the normal to the curve $$x=a\cos t+at\sin t, y=a\sin t-at\cos t$$ and the origin. Equation is given in parameterized form. My attempt ...
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Question on circles…

If three circles with radii ${3}$,${4}$,${5}$ touch each other externally at points P,Q and R,then the CIRCUMRADIUS of ∆PQR is...?? My attempt i think that the let the point of the common ...
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Can the boy escape the teacher for a regular $n$-gon?

This is related to Prove that the boy cannot escape the teacher Suppose there is a boy in the center of a regular $n$-gon. The teacher is on the edge of the $n$-gon (but cannot leave the edge) and ...
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1answer
15 views

Normal vector between two parallel lines [closed]

Is there a way to calculate the normal vector of two parallel lines, without calculating the length or the points?
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71 views

Will the boy outwit the teacher in this way? [duplicate]

In the book, Solving Mathematical Problems: A personal perspective (written by Terry Tao), he discusses a problem named (on Analytic Geometry Chapter, page 79): Problem 5.4 (Taylor 1989, p. 34, ...