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

learn more… | top users | synonyms (1)

0
votes
0answers
17 views

Fit an ellipse with known semi-major-axis and points

In my particular case I am given a projection of a circle onto the $xy$-plane and the radius $r$ of said circle. This results in an ellipse with semi-major axis $a$ equal to $r$. Like in this other ...
1
vote
0answers
17 views

$ABCD$ has area $9$. $M$ is in the middle of $AB$ and the edge $BF$ of length $2$ forms an angle of $60º$, Calculate $[CM,CB,BF]$.

$ABCD$ has area $9$. $M$ is in the middle of $AB$ and the edge $BF$ of length $2$ forms an angle of $60º$. Calculate $[CM,CB,BF]$, knowing that $\mathbb{V}^3$ is oriented by a positive basis. ...
1
vote
1answer
34 views

Prove that $x^2-y^2+xy-1=0$ is a ruled surface

I am studying for an analytic geometry, final but I am totally lost for this problem... We didn't even cover this topic in class (my prof didn't show up for class for two weeks) and I have no clue on ...
1
vote
1answer
57 views

Find the equation of a cylinder

Find the equation of the cylinder that has directrix the curve: $x(t)=t, y(t)=t^2/2, z(t)=0$ and the generatrix is parallel to the line $${x-1\over 1}={y+2\over 1}={z\over 3}$$ I would really ...
0
votes
1answer
44 views

What is wrong with my solution to this problem?

The base $ABCD$ of the figure has area $9$. The point $M$ divides the segment $AB$ on ratio $2$ and the edge $BF$ of length $2$ forms an angle of $60º$. Calculate $[CM,CB,BF]$, knowing that ...
6
votes
4answers
290 views

Coordinates of the center of the circle

I am stuck on this problem: If the lines $y=x+\sqrt{2}$ and $y=x-2\sqrt{2}$ are two tangents of a circle and $(0,\sqrt{2})$ lies on this circle then what is the equation of the circle? I ...
1
vote
0answers
16 views

Analytic structures on $S^1$|

I am currently studying Haefliger's paper "Homotopy and Integrablity". During the last chapter, he applies his theory of $\Gamma$-structures to analytic codimension $1$ foliations. Throughout the ...
-3
votes
0answers
30 views

minimal volume of pyramid x0,y0,z0 [on hold]

I dont know how should i even start. I tried to think about something but get nothing. can someone help me please? Thanks
0
votes
2answers
45 views

Finding the equation of a circle through 3 points under given conditions.

This question has me stuck at the very beginning and I dont understand what to do. Dont need the solution, just a hint on what to do. Q.A and B are points in the xy plane, which are 2sqrt2 units apart ...
3
votes
2answers
33 views

How to find the equation of diameter of a circle that passes through the origin?

So this was a question that I was solving that got me stuck. Its as follows: Q. Find equation of diameter of the circle $x^2 + y^2 - 6x + 2y = 0$ which passes through the origin. Now I have tried the ...
5
votes
1answer
40 views

What does a linear equation with more than 2 variables represent?

A linear equation with 2 variables, say $Ax+By+C = 0$, represents a line on a plane but what does a linear equation with 3 variables $Ax+By+Dz+c=0$ represent? A line in space, or something else? On ...
-3
votes
2answers
31 views

Find the equation of the sphere [on hold]

Find the equation of the sphere which passes through the point $(0,3,-4)$ if the tangent plane is $2x + y + 2z = 9$ at the point $(2,1,2)$.
0
votes
1answer
34 views

vertices of a hyperbola the silliest question ever

I'm given that the center of the hyperbola is $(2,1)$ and $a=3$ and asked to find the vertices. Since vertices are on the same line with the axis of symmetry I thought the coordinates should be $(2,1 ...
4
votes
2answers
58 views

The lines $x+2y+3=0$ , $x+2y-7=0$ and $2x-y+4=0$ are sides of a square. Equation of the remaining side is?

I found out the area between parallel lines as $ \frac{10}{\sqrt{5}} $ and then I used $ \frac{|\lambda - 4|}{\sqrt{5}} = \frac{10}{\sqrt{5}} $ to get the values as $-6$ and $14$ . I am getting the ...
0
votes
0answers
18 views

What's the relation between 2 points from 2 different planes?

I'm trying to find the relation between my "text" objects, and my "world" objects. This may be related to development, but I thought this question was better fit for this exchange. I have two ...
-4
votes
1answer
16 views

Find the equations of the tangent and normal to the ellipse [closed]

Find the equations of the tangent and normal to the ellipse $16x^2 + 25y^2 = 400$ at $t = \frac{1}{\sqrt 3}$
2
votes
5answers
362 views

Creative way to find this area

Let's say We have a circle with center at $(0,0)$ with radius $r$ and we have the line $y=a$ where $0 \leq a \leq r$. the question is what is the area that between the circle and the line $y=a$(the ...
1
vote
1answer
29 views

Given the incentre of $\Delta ABC$ and the equations of the angle bisectors what is the locus of the centroid of the triangle $ABC$?

I got this problem on a test yesterday Consider $\Delta ABC$ with incenter $I(1,0)$. Equations of the straight lines $AI$, $BI$, and $CI$ are $x=1$, $y+1=x$ and $x+3y=1$ respectively and $\cot \left( ...
1
vote
0answers
19 views

Intersection of symmetric lines.

So I have to determine if these 2 symmetric lines intersect. I converted them to parametric: $$\begin{align} -6+2t&=10+4s\\ -4+3t&=4-2s\\ -1+2t&=-1-4s \end{align}$$ Now, I know I have ...
3
votes
0answers
21 views

On the solutions of a system of inequalities avoiding Helly's theorem

Let $a_1,b_1,\cdots,a_4,b_4\in\mathbb{R},r_1,\cdots,r_4\in(0,+\infty)$. Show that, if $\not\exists (x,y)\in\mathbb{R}^2$ such that $$ \begin{cases} (x-a_1)^2+(y-b_1)^2\le r_1\\ ...
-1
votes
2answers
58 views

Compute Speed of an object when moving on a circular arc? [closed]

Consider the following figure: An iPhon is moving on a circular arc from point A to point B. The radius of the orbit is f. Consider the case that a men stands and holding his arm horizontal to the ...
1
vote
1answer
22 views

Finding the equation of the new plane after the original has been rotated by an angle

Find the equation of the plane obtained after rotating the plane $x+y+z=1$ by $90^{\circ}$ about its line of intersection with the plane $x-2y+3z=0$. Since I had to choose one of the four given ...
10
votes
1answer
93 views

Something Isn't Right With My Parking

A few days ago in my Calculus BC class we were given a page of 6 challenging end of the year problems. That was a refreshing change from the drudgery we usually do (WebAssign). One of them went like ...
3
votes
2answers
83 views

How to find center of a conic section from the equation?

If we are given a curve in the form $$ax^2+2bxy+cy^2+2dx+2ey+f=0$$ and the following determinant $$\delta=\begin{vmatrix}a&b\\b&c\end{vmatrix}=ac-b^2$$ is non-zero, then this is either a curve ...
0
votes
2answers
41 views

Find centre of circle with equation of tangent given

(4,1) is a point on one end of the diameter of a circle and the tangent through the other end of the diameter has equation 3 x- y=1. Determine the coordinates of the center of circle. What got me ...
0
votes
1answer
34 views

On the definition of sphere in analytic geometry…

Last year, when I was teaching mathematics (analytic geometry) for one of my clever freands, I arrived to the definition of sphere. I said Fix $r>0$, An sphere is the set of all triples ...
2
votes
2answers
100 views

The Efficiency of Random Parking Problem

A few days ago in my Calculus BC class we were given a page of 6 challenging end of the year problems. That was a refreshing change from the drudgery we usually do (WebAssign). One of them went like ...
1
vote
2answers
23 views

Find the slope of a line $L$ that tangent to the graph of $y = x^3$ and passes through the point $(0,2000)$.

Find the slope of a line $L$ that tangent to the graph of $y = x^3$ and passes through the point $(0,2000)$. Well, I am new to this concept, to me, slope means $\dfrac{dy}{dx}$, but I get ...
3
votes
1answer
106 views

Largest of the smallest angles of incidence from arbitrary point to tetrahedron vertex/centroid line

Picture a regular tetrahedron where each vertex has a line through the centroid and a plane normal to it. I need to show that the range of the smallest angles of incidence from an arbitrary point to ...
1
vote
2answers
53 views

Let $y=x^2+ax+b$ cuts the coordinate axes at three distinct points. Show that the circle passing through these 3 points also passes through $(0,1)$.

Let $y=x^2+ax+b$ be a parabola that cuts the coordinate axes at three distinct points. Show that the circle passing through these three points also passes through $(0,1)$. Since, the graph of the ...
1
vote
1answer
37 views

Conjugate Hyperbolas.

What would be a good approach to tackle this problem. In a previous assignment I managed to show Pq=Pr. How do I show that this tangent intersects the conjugate hyperbola. Should I start by ...
0
votes
1answer
26 views

Analytic geometry, distances

Find the equation of the geometric place: Whose distance to the point $(4,0)$ equals half the distance to the straight line $x=19$ Im using the formula for distance between points $P(4,0), Q(19,0)$ ...
2
votes
4answers
35 views

Finding horizontal tangents to a function.

Find the points at which the line tangent to the following function is horizontal $$q(x)=(x+3)^4(2x-1)^7$$ Every time I've gotten to the point of finding $x$ the numbers are all irrationally too ...
0
votes
0answers
30 views

Reference request: history of analytic geometry

I am searching a book in the domain of the history of math, that describes the historical origins of analytic geometry, starting from Descartes (?), and that describes also its development (e.g. the ...
0
votes
0answers
8 views

analytic geometry question involving perpendicular vectors

Determine the parametrics equations of the straight line that passes by $A(-1,4,5)$ and is perpendicular to $r:P=(-2,1,1)+t(1,-1,1)$. Someone can solve this? I'm trying for more than a hour and I'm ...
0
votes
0answers
25 views

Intersection of a curve with a complex line

Given: $$ \left\{\begin{matrix}t =\frac{1}{n}\sqrt{n^{4}-z^{2} } & \\ z=im & \end{matrix}\right.$$ with $n<m$, positive integers (and $i$ the imaginary unit), if one wanted to ...
1
vote
1answer
29 views

Definition of angle between non-differentiable curves

(Background: I am trying to understand the definition of angle-preserving function..I posted a question earlier but I still have doubts) My question is:how is the angle between two curves defined if ...
0
votes
0answers
31 views

Using axis coordination to represent rotation matrix instead of angles

Euler angles give us clear matrix for conversion of a vector from car reference $Fr^C$ to earth reference $Fr^E$. If $\vec V$ is a vector in different frames it is represented differently: $$\vec ...
0
votes
2answers
18 views

Analytic geometry and definite integrals problem…

So, here's the problem: We have a parabola $y^2=2px$ and a line which is perpendicular to parabola and forms the angle $\frac{3\pi}{4}$ with x axis. I have to find the area between the parabola and ...
1
vote
0answers
26 views

Given five points and a line find the points of the line that lie in the conic through the five points [closed]

So I'm given 5 points in general position and a line, I already know the method using Pascal's theorem to find points in the conic but I dont know how to find specifically the ones that lie on a given ...
1
vote
2answers
62 views

Easiest way to verify that $4x^2+y^2=1$ is an ellipse?

Normally I would just divide both sides by the number $4$ because it's not good in there, but I can't do it for $$4x^2+y^2=1$$ I must have $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ So what's the ...
0
votes
2answers
20 views

Characterize a rotation matrix

Given a matrix $A\in M_{2 \times 2}(\mathbb R)$ or $M_{3\times 3}(\mathbb R)$ how to determine if it is a rotation matrix? Is there any theorem that characterize a rotation matrix just by looking at ...
5
votes
1answer
55 views

Is there a problem in assuming that a point is the same thing of a vector?

I've read Apostol's Calculus, in the section on analytic geometry. He says that he's going to use 'vector' and 'point' interchangeably. But in Beardon's Algebra and Geometry, he argues that there is ...
0
votes
3answers
56 views

Coordinate Geometry - Area of a Quadrilateral

What is the area in square units, of a quadrilateral whose vertices are $(5,3), (6,-4), (-3,-2), (-4,7)$ ? I have tried creating the triangles, but didn't know how to find the diagonal. I wanted to ...
3
votes
2answers
111 views

snugly fitted spheres in a cube

A larger sphere A, having a radius $R$ is snugly fitted in a cube (i.e. sphere A touches all six faces of the cube). Further, a small sphere B is snugly fitted in the corner of cube (i.e. sphere B ...
1
vote
0answers
33 views

Analytic Geometry - vectors and points

Can somebody help me? In the picture, $\|AM\|=2\|MB\|$ and $\|AN\|=\frac{1}{3}\|CN\|$. Write $X$ in function of $A, AB, AC$.
1
vote
1answer
36 views

Hyperplanes divide space

Problem. What is maximal number of connected components on which $n$ hyperplanes divides $\mathbb{R}^m$ if they all have 1 common point. In fact this problem was firstly stated in $\mathbb{R}^3$ and ...
1
vote
2answers
52 views

Surface area of a section of the unit sphere

Let $v$ be a vector on the unit sphere in $\mathbb{R}^n$ and let $S(\epsilon)$ be the set of vectors $s$ on the same sphere such that $$ |s \cdot v| \leq \epsilon.$$ What is the surface area of ...
0
votes
1answer
51 views

High School Geometry problem with a triangle and trapezoid in a larger triangle.

In school, I have an assignment to write a problem for geometry students. I have written the following problem. Draw triangle ABC. Let the height have magnitude h. Draw a line segment, DE, which is ...
0
votes
1answer
25 views

Determine the value of y so that two line segments are parallel

Determine the value of $y$ so that the line segment with endpoints $P(3, y)$ and $Q(-3, -1)$ is parallel to the line segment with endpoints $R(-4, 9)$ and $S(5,6)$. I began by finding the slope ...