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

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Problem solving on Co-ordinate Geometry.

Two fixed straight line $OX$ and $OY$ are cut by a variable line in the points $A$ and $B$ respectively and $P$ and $Q$ are the feet of the perpendiculars drawn from $A$ and $B$ upon the lines $OBY$ ...
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1answer
35 views

Curve such that if $P = (x_0,y_0)$ lies on $C$, then $P$ is the midpoint of the tangent line to $C$ at $P$ contained in the first quadrant.

Find a curve $C$ in the first quadrant in $\mathbb{R}^2$ passing through $(3,2)$ with the property that if $P = (x_0,y_0)$ lies on $C$, then $P$ is the midpoint of the tangent line to $C$ at $P$ ...
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1answer
30 views

Books about the general equation of the quadrics

I'd like suggestions of books that address the general equation of the quadrics ($Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0$), that is, a book that teach to rotate a quadric and also ...
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1answer
23 views

line of symmetry of a square

The problem goes; The entire figure is placed in the standard $(x,y)$ coordinate plane such that the vertices of the square are $A(6,6)$, $B(0,6)$, $C(0,0)$ and $D(6,0)$. The $x$ coordinate of $E$ is $...
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1answer
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Find the equations of the tangent planes to the sphere $x^2+y^2+z^2+2x-4y+6z-7=0,$ which intersect in the line $6x-3y-23=0=3z+2.$

Find the equations of the tangent planes to the sphere $x^2+y^2+z^2+2x-4y+6z-7=0,$ which intersect in the line $6x-3y-23=0=3z+2.$ Let the tangent planes be $A_1x+B_1y+C_1z+D_1=0$ and $A_2x+B_2y+...
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122 views

Averaging transformation of a closed plane curve

Let's suppose we have a closed plane curve of some shape whose points are described by the single parametric equation $P(x(t), y(t))$ where $t$ is some increasing parameter (example circle) or by ...
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1answer
40 views

Given list of points, find lines with more than two points

I recently was asked to come up with an algorithm to find all the lines for the given set of points that have more than two points on them. For instance if I have: ...
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1answer
35 views

Prove that as $PP'$ varies,the circle generates the surface $(x^2+y^2+z^2)(\frac{x^2}{a^2}+\frac{y^2}{b^2})=x^2+y^2.$

$POP'$ is a variable diameter of the ellipse $z=0,\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ and a circle is described in the plane $PP'ZZ'$ on $PP'$ as diameter.Prove that as $PP'$ varies,the circle ...
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1answer
74 views

Locus of the center of the circle of radius $a$,which always intersects coordinate axes

If the axes are rectangular, show that the locus of the center of the circle of radius $a$,which always intersects coordinate axes is $x\sqrt{a^2-y^2-z^2}+y\sqrt{a^2-z^2-x^2}+z\sqrt{a^2-x^2-y^2}=a^2$ ...
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1answer
113 views

Show that any smooth projective curve of genus zero over a field $K$ is isomorphic to a plane conic over $K$

I have the following question: Show that any smooth projective curve of genus zero over a field $K$ is isomorphic to a plane conic over $K$. Assuming that a plane conic is a conic cut by a plane,...
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1answer
37 views

Largest equation of a circle that shares 2 tangents with a curve

Just played around on a graphic calculator a little, and discovered that given the curve $y=x^2$ , all circles with the equations in the form of $\left(y-a\right)^2+x^2=\frac{4a-1}{4}$ for all $a>0....
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4answers
65 views

Find the center of the circle through the points $(-1,0,0),(0,2,0),(0,0,3).$

Find the center of the circle through the points $(-1,0,0),(0,2,0),(0,0,3).$ Let the circle passes through the sphere $x^2+y^2+z^2+2ux+2vy+2wz+d=0$ and the plane $Ax+By+Cz+D=0$ So the equation of ...
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1answer
53 views

Maximium and minimum value of area.

Given that the equation of parabola is $y=x^2+1,1\leq x\leq 3$ What is the maximum and minimum value of area formed by x-axis,tangent,normal at any point on parabola. Now I wrote the equation as $x^2=...
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1answer
11 views

Find the equation of the sphere $OABC.$

$OA,OB,OC$ are mutually perpendicular lines through the origin and their direction cosines are $l_1,m_1,n_1;l_2,m_2,n_2;l_3,m_3,n_3.$If $OA=a,OB=b,OC=c,$prove that the equation of the sphere $OABC$ is ...
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1answer
63 views

Finding minimum difference between two linear functions

Given two functions of the form $y = m_1x + c_1$ and $y = m_2x + c_2$ where $m_1,m_2,c_1,c_2$ are positive integers. How to find the absolute minimum difference between the two functions for positive ...
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0answers
14 views

Show that the center of the sphere lies on the line $z=0,x^2+y^2=(a^2-c^2)\csc^22\alpha$

A variable sphere passes through the points $(0,0,\pm c)$ and cuts the lines $y=x\tan\alpha$, $z=c$; $y=-x\tan\alpha$, $z=-c$ in the points $P,P'$. If $PP'$ has constant length $2a$ show that the ...
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Line-Torus intersection

I have a vector equation for a line given by $\mathbf r = \mathbf a +\mathbf bt$. I would like to find the intersections that it makes with a torus given by $$\left(\mathbf r \cdot \mathbf r - R^2 - ...
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3answers
51 views

Obtain the equation of the sphere which passes through the points $(1,0,0),(0,1,0),(0,0,1)$ and has its radius as small as possible.

Obtain the equation of the sphere which passes through the points $(1,0,0),(0,1,0),(0,0,1)$ and has its radius as small as possible. Let the sphere passes through $(x_1,y_1,z_1)$ Then i obtained ...
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2answers
19 views

Maximum distance from closest vertex of rhombus

Consider the unit rhombus formed by joining following coordinates $A(0,0), B(1,0), C(\frac{3}{2}, \frac{\sqrt{3}}{2}), D(\frac{1}{2}, \frac{\sqrt{3}}{2})$ What is the largest possible distance from ...
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Domain of validity of a certain inequality with 2 variables

For which values of x, y does the equality $$x^2y^2+x^2+y^2+4 \leq 6xy$$ hold ? Please could you help with this problem as I am having trouble getting started.
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2answers
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Equation of Circle

Prove that the equation $x^2+y^2+2gx+2fy+c=0$ always represents a circle. I just don't have any idea regarding this. Can anyone help me? Help much appreciated! Thanks..
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1answer
79 views

Intersection of plane

How would I do this question. Find a plane that contains the point A(3,1,−1) and touches the cylinder with radius 3 whose axis is the line p : x = 0, y = z. It was on a test I did a few days ago and ...
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0answers
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If one of the focii of ellipse is moved to infinity, then how it becomes a parabola?

Eccentricity of Parabola is 1. Eccentricity of ellipse is <1. i.e. FP / PM = 1 for parabola. Then how an ellipse will become a parabola if F1P / PM =e<1 if F1 is any one of its focus?
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Logarithmic Spiral

Let's assume a particle starts on an equilateral triangle of side length "A" with some constant speed u. The particle goes on a logarithmic spiral around the centroid. Find the distance covered by the ...
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1answer
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One side of a triangle is the geometric mean of the radii of the inscribed and circumscribed circles

Let $ABC$ be a triangle, $\alpha$ the angle adjacent to point $A$, and $a$ the opposite side. If $a$ is the geometric mean of the radii of the inscribed circle $r$ and circumscribed circle $R$, how ...
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1answer
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How to show that a tangent line to a parabola is a perpendicular bisector

Let $P$ be the parabola given by the equation $P(x)=\frac{1}{4}x^2$ and $a\in\mathbf{R}$. Let $F=(0,1)$ be the focus of $P$ and $R=(a,P(a))$ be a point on $P$ and let be the point $V=(a,-1)$. I want ...
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0answers
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Bending a horizontal from 0 to infinity real number line, ninety degress counter-clickwise at 1.

Can the real number line from 0 to infinity, which of course is often represented as a horizontal straight line, also be represented as being bent ninety degrees counter-clockwise at 1? I.e., if such ...
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1answer
36 views

(Geometric) Intuition behind Different Types of Rank 2 Tensor (Specifically Quadratic Forms)

This is essentially a follow-up to this question: Differences between a matrix and a tensor I think I have a good intuition/idea for the change of basis for a rank-(1,1) tensor ($A\vec{v} = \vec{w}$) ...
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2answers
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Condition for three points to lie on a Sphere?

If A and B are the points (3,4,5) and (-1,3,-7) respectively then the set of the points P such that $PA^2+PB^2 = K^2$ where K is a constant lie on a proper sphere if K = 1 or K^2 <>= 161/2? The ...
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1answer
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mathematics was recreated on a foundation of number concepts rather than geometrical ones

In Richard Courant and Fritz John's book Introduction to Calculus and Analysis Volume I, says In modern times mathematics was recreated and vastly expanded on a foundation of number ...
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1answer
38 views

Is there a simpler matrix for these rotation?

I'm still in high school so sorry if I do not know this. I learnt matrices in class and how to use them to rotate by $90^o,180^o,$ and $270^o$ with center (0,0). I played around with them later and ...
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Congruent number $23$

The set of Congruent numbers are all the integer areas of rational sided right triangles. This means that if g is a Congruent number there exists some integer $n$ such that $g \cdot n^2$ is the ...
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1answer
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The lenght of rectifiable curve in $\mathbb{R}^n \setminus B[0,r]$ that connects antipodes points.

This question is from my homework, here it goes: Let $\gamma \colon [a,b] \to \mathbb{R}^n \setminus B[0,r]$ be a rectifiable curve such that $\gamma(a)=-\gamma(b)$. Using euclidean norm prove that $...
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4answers
96 views

Find the minimum distance to move an ellipse to be inside another ellipse?

For the problem of ellipse intersection, I would like to know an accurate "general, including the cases of two non intersected ellipses, and non aligned ellipses" method to calculate the minimum ...
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2answers
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Show that there are at most two rational points on $(x - a)^2 + (y - b)^2 = r^2$ for $a, b$ irrational.

For any given irrational numbers $a, b$ and real number $r \gt 0$, show that there are at most two rational points (points whose coordinates are both rational numbers) on the circle $(x - a)^2 + (y - ...
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1answer
10 views

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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28 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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Parabola, tangent and angles (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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1answer
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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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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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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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13 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
20 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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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
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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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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
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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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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
40 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 (...