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

learn more… | top users | synonyms (1)

2
votes
2answers
47 views

Find the line segments cut off by the plane $ax+by+cz+d=0$ on the coordinate axes, if $abcd\neq 0$

I'm reading Pogorelov's Geometry. Find the line segments cut off by the plane $ax+by+cz+d=0$ on the coordinate axes, if $abcd\neq 0$. Writing the equation as $a(x-x_0)+b(y-y_0)+c(z-z_0)=0$, I ...
2
votes
1answer
54 views

What is the need to define so many forms of equation of a straight line?

When I study maths, I try to understand why the mathematicians brought out this concept or what usefulness they might have seen in the concept that they worked upon. So when I started with straight ...
3
votes
1answer
85 views

Surface Area of unit n-sphere covered by rotating a unit vector around a fixed unit vector such that angle between the two vectors is always fixed.

Consider an n-dimensional unit sphere and unit vector from the origin with its tip lying on the surface of sphere. Consider another vector which makes some angle say $\epsilon$ with unit vector. From ...
2
votes
1answer
44 views

Placing $n$ points so that their distances lie in $[1,a]$

What is the maximum number of points we can place in the plane so that the distance between any two such points is in the interval $[1,a]$? I had initially conjectured that the maximum could be ...
0
votes
2answers
531 views

Find the equation of circle touching given lines and a given point. [closed]

$U: 3x+4y-20=0$ and $v:3x+4y+10=0$ are two straight lines. Find the equation of circles touching the given lines and passing through point $P(1,2)$.
0
votes
1answer
89 views

Equation of the affine transformation that fixates a certain line

I have to find the equation of the affine transformation of the affine plane $A_2$ that (1) fixates the line $s: x + y - 1 = 0$ and (2) such that $A(Q)=P$, where $Q(1,2)$ and $P(2,1)$. How should I ...
0
votes
1answer
23 views

Question about determinig types of surfaces?

$$x^2 +y^2 +z^2 +2x +1=0$$ This is an equation for dot if we are talking about surfaces, right? It is not an ellipsoid.
0
votes
1answer
53 views

How many ellipsoids can be maximally inside a circle?

This discussion is related to this discussion here where I want to deduce the area difference between such two circles filled with ellipsoids. Actually, to understand this difference is the main ...
7
votes
2answers
154 views

How to determine whether a point is inside a closed region or not?

Take the following parametric equation of an implicit curve as an example: $$ \left\{\quad \begin{array}{rl} x=& 9 \sin 2 t+5 \sin 3 t \\ y=& 9 \cos 2 t-5 \cos 3 t \\ \end{array} \right. $$ ...
2
votes
0answers
31 views

Finding the transformation matrix of a projective transformation in RP^2

So I want to understand how to find the matrix that represents the projective transformation that sends 4 given points to 4 given images, I know that 4 points are necessary to determine it but I can't ...
1
vote
2answers
188 views

Bisector of two lines in the euclidean space $\mathbb{E}_3$

Let $$r: \begin{cases} x + z = 0 \\ y + z + 1 = 0\end{cases}$$ and $$s: \begin{cases} x - y - 1 = 0 \\ 2x - z -1 = 0\end{cases}$$ be two lines in the euclidean space $\mathbb{E}_3$. It is easily ...
1
vote
1answer
47 views

Verifying if these basis are positive or negative?

Verify if the basis $E=(e_1,e_2,e_3)$ and $F=(f_1,f_2,f_3)$ are positive or negative with: $$f_1=e_1\quad \quad\quad\quad\quad f_2=e_2+e_3\quad \quad \quad\quad \quad f_3=e_1+e_2 $$ I did the ...
1
vote
1answer
64 views

Finding the smallest square inside a parabola. [duplicate]

I just thought of a problem earlier today, but wanted to know if there was an easier way of acquiring the answer. Say I have a standard parabola $y=x^2$ with 3 points on it $P,Q,R$ and another point $...
0
votes
1answer
36 views

Geometric proof that (symmetry w/r to $x$ and $y$ axes) $\implies$ (symmetry w/r to origin)

I'm trying to prove that reflecting a point about the x and y axes is equivalent to reflecting it about the origin. Is my proof valid? How could I improve it? Proof: Take a point $a$ in the first ...
1
vote
3answers
62 views

Breaking down the equation of a plane

Could someone explain the individual parts of a plane equation? For example: $3x + y + z = 7$ When I see this I can't imagine what it's supposed to look like.
0
votes
0answers
75 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
27 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
48 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
137 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
56 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 $\mathbb{V}...
6
votes
4answers
780 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
21 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 ...
0
votes
2answers
71 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 ...
4
votes
2answers
519 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 ...
6
votes
1answer
59 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 ...
4
votes
2answers
59 views

The four straight lines given by the equation $12x^2+7xy-12y^2 =0$ and $12x^2+7xy-12y^2-x+7y-1=0$ lie along the side of the?

I know these equations are called general equation of second degree and also represent a pair of straight lines. I could extract lines from the equation $$12x^2+7xy-12y^2 =0 $$ (these are $$ 3x+4y=0$$ ...
0
votes
1answer
41 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
68 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
20 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 ...
2
votes
3answers
50 views

how to prove by contradiction that any distance between a curve $x^4 - x^2 + y^4 - y^2 = 0$ and the origin is less than or equal to $\sqrt{2}$

Given a closed trajectory $x^4 - x^2 + y^4 - y^2= 0$ Prove that any distance between any point on the curve and the origin does not exceed $\sqrt2$ (ie, maximum distance from the origin to the curve ...
2
votes
5answers
384 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
45 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
50 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
29 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\\ (x-a_2)^2+(y-b_2)...
1
vote
1answer
130 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
112 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 ...
5
votes
3answers
330 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
697 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 $3x- y=1$. Determine the coordinates of the center of circle. What got ...
1
vote
1answer
63 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 $(x,y,z)...
2
votes
2answers
151 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
35 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 $\dfrac{...
0
votes
1answer
105 views

% of traversal for a point between two other points along a line

I'd like to solve for some "phase" or percentage, involving an arbitrary location (xC, yC) between two points. I'm not familiar with how to phrase this question, so please excuse my ignorance. Not ...
3
votes
1answer
118 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
90 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
49 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
27 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
62 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
57 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
12 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
1answer
22 views

Find correleation between values and degrees

I have an arc that starts at $252$ degrees and ends at $288$ degrees, I would like to assign non - linear values on it with this ratio: $1 - 180$ degrees. $5 - 135$ degrees. $10 - 90$ degrees. $30 - ...