# Tagged Questions

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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### 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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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Мöbius Transformations and circle inversion

Can a Möbius Transformation be decomposed into a composition of 2 generalized circle inversions?
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### 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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### 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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### 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 ...
### 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 ...