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

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1answer
24 views

Appolonius three circles problem, finding centre of tangent circle (analytic geometry)

Given 3 circles: $C_1$ centered at $(0,0)$ with radius 1 $C_2$ centered at $(a,0)$ with radius $a+1$ $C_3$ centered at $(-a,0)$ with radius $a+1$ (so $C_1$ is internaly tangent to both $C_2$ and ...
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0answers
15 views

Surface of a torus in terms of Legendre polynomials

The equation of a spheroid is $$\frac{x^2 + y^2}{a^2} + \frac{z^2}{b^2}$$ Its surface can be expressed as $$ r = a \left( 1 - \frac{2}{3} \epsilon P_2(\cos \theta) \right) $$ where $r$ is the ...
3
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1answer
41 views

Least value of $(α-β)$ if area is minimum.

The area of parallelogram formed by the lines $$x \cos \alpha +y\sin\alpha = p,$$ $$x \cos\alpha +y\sin\alpha = q,$$ $$x\cos\beta + y\sin\beta = r$$ and $$x\cos\beta + y\sin\beta = s$$ for given ...
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2answers
31 views

Calculus: Find equation that represents the set of all points that are equidistant from given three points (0,0,0) (2,4,3)(10,8,9)

So it is essential circumcenter problem in 3D that involves multivariable calculus. If you could at least help with the ideas or steps of tackling this problem that would be great PS: I made up the ...
2
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3answers
31 views

Circle of radius of Intersection of Plane and Sphere

The plane $x+2y-z=4$ cuts the sphere $x^2+y^2+z^2-x+z-2=0$ in a circle of radius? I tried putting value of y from plane in sphere but then I get a $zx$ term. How to proceed?
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1answer
48 views

What is the length of [BC]?

Let A , B and C be 3 points of a circle (c) P is the intersection of two tangents of the circle in points B and C Let (AB)//(CP) and AB=3 and BP=4 What is the length of BC Can someone give hint ! ...
3
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1answer
51 views

What is the Newton's general theory of diameters?

I was reading a book on Mathematics, which contained this topic. I was not able to grasp the concept. There was not much info on internet also. It was as follows: Let an $n$th order curve be ...
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3answers
43 views

Distance between point and line in point slope form on a plane

If I have an equation in point slope form $$y=mx$$ how can I use the perpendicular distance formula: $$\text{Perpendicular Distance} = \frac{\left | Ax_{1} + By_{1} + C\right |}{\sqrt{A^2 + B^2} }$$ ...
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2answers
52 views

Formula that describes the movement of a bishop in chess

I'm programming a chess game and I'm trying to validate the movements every player tries to make. Obviously, every piece can move differently and I've had no trouble validating their moves up until ...
3
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2answers
41 views

Polar radius of a general ellipsoid

Is there a proper parametrization of a general ellipsoid in spherical coordinates? The regular parametrization is this: $$x=a\cdot \cos\phi \cos\theta\\y=b\cdot \cos\phi \sin \theta\\z=c\cdot ...
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0answers
38 views

Hyperbola and 3 normals from point P

From any point P on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ three normals other than that at P are drawn. Find the locus of the centroid of the triangle formed by feet of the normals. Do we ...
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1answer
53 views

Is $f(x+a) - f(a) = f(x) + f'(a) x$ an identity?

Given a differentiable function of $x$, denoted by $f(x)$; is $f(x+a) - f(a) = f(x) + f'(a) x$ an identity? For example, if $f(x)=x^2$, then it gives $(x+a)^2 - a^2 = x^2 + 2ax$, which is true. So, ...
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1answer
27 views

Image of a circle under conformal map $1/z$

The image of a circle under conformal map $1/z$ should be a circle, but how to prove it (or how to find the relationship between the two circles)? $z = x + iy = d + a\exp(i\theta)$, where $a$ is the ...
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1answer
16 views

Find the equations of Lines given Points and Angle

I have the following scenario. The coordinates of points B and D are $(10,0)$ and $(10,-10)$ respectively. I want to construct angle $\angle BFD = 45^\circ $. How can i find the coordinates of ...
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2answers
22 views

finding the coordinate given a distance with its coordinates

A point $P(x,y)$ has a distance $5\sqrt{2}$ units from $Q(4, -7)$ and a distance $\sqrt{106}$ units from $R(-6,5)$. Knowing that, find $P$. the image is exactly the set of problem that our professor ...
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0answers
21 views

Finding normal of N coplanar points

I am programming a video game vehicle physics simulation. I am currently using the method described in the post below. To ordinate the car body rotation Computing the best-fit plane normal from n ...
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1answer
24 views

Number of places two intersecting lines can intersect a hyperbola

If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola,then what are the possible numberof places where the lines can intersec the hyperbola ? ...
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2answers
25 views

How to rotate an hyperbola by $45^\circ$ so that I have an equation of the form $xy=c$

I am trying to show that If I rotate an hyperbola of the form $\cfrac{x^2}{a^2}-\cfrac{y^2}{b^2}=1$ by $45^\circ$ I get an equation of the form $x'y'=c$. Using the following rotation coordinates: ...
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0answers
22 views

How to get rotation coordinates of point $(x',y')$ in terms of $(x,y)$

Given that we have $x=x' \cos \theta - y' \sin \theta $ and $y=x' \sin \theta +y' \cos \theta $ ,how can I express $x',y'$ in terms of $x,y$ and $a$ ? I've browsed through the site to seek for some ...
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4answers
497 views

Prove that the boy cannot escape the teacher

I'm struggling with the following problem from Terence Tao's "Solving Mathematical Problems": Suppose the teacher can run six times as fast as the boy can swim. Now show that the boy cannot ...
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1answer
34 views

Shrinking a rectangle enscribed about another, such that it is enscribed by the other

I ran into this problem during my work, and I've been banging my head against it for a bit. Seems like a simple enough algebra problem, but I'm thinking it'll need to be solved numerically. Here's ...
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0answers
23 views

Equation curves

Introduce equation curves to the canonical form, finding an appropriate rectangular coordinate system. a) $5x^2+12xy-22x-12y-19=0$ b) $9x^2+24xy+16y^2-230x+110y-475=0$ Could somebody do one task. I ...
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1answer
41 views

Find the least eccentricity of an ellipse which can rest on the plane.

A perfectly rough plane is inclined at an angle $\alpha$ to the horizontal.Find the least eccentricity of an ellipse which can rest on the plane. Any ideas on how to solve this problem?
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1answer
45 views

How to find a point in a Right Triangle given 2 known points, all sides, all interior angles

This triangle is not parallel or vertical, it's in a 2d plane. Distance formula gave me very troubling results, looking to use SOH CAH TOA, particularly a simple method and not a complex method of ...
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2answers
81 views

Calculate Third Point of Triangle

I'm trying to calculate the third point of a triangle: I know two points, (2,3) and (5,2) and the angles at this sides, both of ...
0
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1answer
21 views

Calculate vector from two points and angle

I have three points: P1 (5065, 423) P2 (4935, 281) P3 (0, 0) I calculated the angle between P3,P2 and ...
0
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1answer
36 views

How do I get a point from two points using a right triangle?

In the image the triangle is made up of 3 points, 2 of which are found, the third one is missing, not sure how to get this last point. [Need an Equation]
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1answer
17 views

What is the easiest way to find the radius and center of the circle of intersection between two spheres?

If given two spheres $S_1$ and $S_2$, of radius $r_1$ and $r_2$, centered at 3-space points $P_1$ and $P_2$, respectively. What is the easiest way to find the radius and center of the circle of ...
0
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1answer
10 views

Sphere centered in line $(x,y,z) = (-2,0,1) + \lambda (0,0,1)$ tangent to planes $ x-10z = 0 $ and $ x+2z = 0 $ whose radius squared is $r^2 > 20$

how may I find the sphere centered in line $$(x,y,z) = (-2,0,1) + \lambda (0,0,1)$$ tangent to planes $$ x-10z = 0 $$ and $$ x+2z = 0 $$ whose radius squared is $$r^2 > 20$$ Thank you. ...
0
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1answer
23 views

Ellipse by moving center of a parametric circle equation?

Given that a parametric eq for a circle is given by : $$x= r \cos \theta \\ y= r \sin \theta $$ Is it possible to move the center of circle by a (periodic) function $f(r,\theta)$: $$\begin{align} x ...
1
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1answer
30 views

Is there an integer point in the intersection of a rational cone and a rational hyperplane?

I'm thinking about a problem: given a cone, generated by some integral vectors $b_1,\cdots,b_n$, which means components of these vectors are integers. And the cone in $\mathbb{R}^n$ is ...
39
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4answers
2k views

Why are we justified in using the real numbers to do geometry?

Context: I'm taking a course in geometry (we see affine, projective, inversive, etc, geometries) in which our basic structure is a vector space, usually $\mathbb{R}^2$. It is very convenient, and also ...
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2answers
57 views

Find $a$, $b$ such that the ellipse $(x/a)^2 + (y/b)^2 = 1$ passes through $(\sqrt 2, 2)$ and has minimum area

I am working on a problem in which, for $a$, $b \gt 0$, we let $(x/a)^2 + (y/b)^2 = 1$ describe an ellipse. I am required to use the method of Lagrange multipliers and the corresponding second ...
0
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1answer
61 views

Get coordinate origin from two points

I want to draw a Image onto a Layer. Where to put this Image I know from two Points A and b: Point A = Image-Cordinates( 4925, 281) Layer-Coordinates...
1
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1answer
34 views

Determine the equation of a transformed elliptical arc

I have difficulties determining the equation of a transformed elliptical arc which my company needs for their laser marking software. I'm starting out with an elliptical arc that is in either ...
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0answers
90 views

Envelope of an Envelope

Background and Motivation Consider the following equation of family of ellipses in polar coordinates $$ r(\theta, \alpha ) = a\;\frac{- e \cos(\theta+ \alpha )+ \sqrt{ 1 - e^2 ...
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4answers
41 views

Distance between points - equation of a line

I have worked on this particular example: The distance between the point $M_1(3,2)$ and $A$ is $2\sqrt5$ and the distance between the point $M_2(-2, 2)$ and $B$ is $\sqrt5$. Come up with a equation ...
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0answers
36 views

What is the polar of an Euclidean unit ball but its center is not in the origin

I know that the polar of an Euclidean unit ball is itself, but I wonder what if its center is not in the origin, like: $$ B=\{(x,y)\in \mathbb R^2 \mid(x-1)^2+y^2\le1\} $$ and the polar of a set is: ...
0
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1answer
18 views

minimization of sum of distances using elementary methods

I want to find a solution to the following minimization problem using only elementary methods. This is to say: high school algebra, basic inequalities, basic trigonometry and trigonometrical ...
1
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1answer
21 views

How did he went from $b_{12}$ to $\tan 2\varphi=\cfrac{2a_{12}}{a_{11}-a_{22}}$?

I'm reading Aarts': Plane and Solid Geometry. Here: How did he go from $b_{12}$ to $\tan 2\varphi=\cfrac{2a_{12}}{a_{11}-a_{22}}$? I have tried a few things on paper but I throw it all away ...
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2answers
41 views

Scale a rectangle from a point other than its center [closed]

How do I scale a rectangle from a point other than the centre of the rectangle? Specifically, I am trying to determine the new X and Y position of a rectangle after having rescaled it, taking into ...
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2answers
24 views

Express the region $D=\{(x,y): x^2\leq y \leq x^2+x^3, x>0 \}$ as the union of cubic curves

Let $D=\{(x,y): x^2\leq y \leq x^2+x^3, x>0 \}$ I know the family of curves $\gamma(t)=x^2+tx^3$ belong to D, for $t\in [0,1]$. It is true that for every $(x,y)\in D$ there exist a unique ...
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5answers
57 views

Find equation of the straight line satisfying some conditions

Can some one please help me to solve the following question. Find equation of the straight line that goes through $(2 , -5 )$ and: has slope $-3$ is parallel to the $x$-axis is parallel to the ...
1
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0answers
34 views

On a non-standard approach to the classification of conics?

I've been introduced to a method of classifying conics but it's too cumbersome for me. I've discovered something that seems a little more promising on Eves' Elementary Matrix Theory: And ...
1
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1answer
28 views

Order of affine reflections (described with complex numbers operations)

Let be the affine reflection described as an operation with complex numbers : $$s_\beta,_\nu : z \mapsto \nu + \overline{\beta z},$$  where $z, \nu \in \Bbb C$ and $\beta \in \Bbb C^1 = \{x+iy \ ...
2
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1answer
31 views

The line $\frac{x+6}{5}=\frac{y+10}{3}=\frac{z+14}{8}$ is the hypotenuse of an isosceles right angled triangle whose opposite vertex is $(7,2,4)$

The line $\dfrac{x+6}{5}=\dfrac{y+10}{3}=\dfrac{z+14}{8}$ is the hypotenuse of an isosceles right angled triangle whose opposite vertex is $(7,2,4)$.Find the equation of the remaining sides. My ...
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0answers
30 views

What does it means to find an solution for an elliptical equation?

What does it means to find integer solutions for $(x-2y)^2+2(y-6)^2=102$ , for example.
2
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0answers
46 views

Prove that planes $AOA',BOB'$ and $COC'$ pass through the line $\frac{x}{l_1+l_2+l_3}=\frac{y}{m_1+m_2+m_3}=\frac{z}{n_1+n_2+n_3}$

$O$ is the origin and lines $OA,OB$ and $OC$ have direction cosines $l_1,m_1,n_1;l_2,m_2,n_2;l_3,m_3,n_3$ respectively.If lines $OA',OB'$ and $OC'$ bisect angles $BOC,COA$ and $AOB$,respectively,prove ...
1
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1answer
26 views

If the projections of $OA$ and $OB$ on the plane $z=0$ make angles $\phi_1$ and $\phi_2$,respectively,with the $x-$axis

$OA,OB,OC,$ with $O$ as origin, are three mutually perpendicular lines whose direction cosines are $l_1,m_1,n_1;l_2,m_2,n_2$ and $l_3,m_3,n_3$ respectively. If the projections of $OA$ and $OB$ on the ...
1
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2answers
33 views

Prove that for all values of $\lambda$ and $\mu,$ the two planes intersect on the same line.

Prove that for all values of $\lambda$ and $\mu,$ the planes $\frac{2x}{a}+\frac{y}{b}+\frac{2z}{c}-1+\lambda(\frac{x}{a}-\frac{2y}{b}-\frac{z}{c}-2)=0$ and ...