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

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Permutation Combination

Let $C = \{(i, j)|i, j \in \mathbb Z,\; 0 ≤ i, j ≤ 24\}$. How many squares can be formed in the plane all of whose vertices are in $C$ and whose sides are parallel to the X−axis and Y − axis?
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1answer
331 views

Angle bisectors for pair of lines

Suppose you have a pair of lines passing through origin, ax^2 + 2hxy +by^2 = 0, how would you find the equation of pair of angle bisectors for this pair of lines. I can do this for 2 separate lines, ...
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1answer
53 views

Coplanar vectors

Prove that if $$\vec{a}\times\vec{b}+\vec{b}\times\vec{c}+\vec{c}\times\vec{a}=0$$ then $\vec{a},\vec{b},\vec{c} $ are coplanars. One thing I know is that i have to get $\vec{a}\cdot(\vec{b}\times\...
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0answers
479 views

Family of circles touching a line

I found this in a book but I am not able to understand how they got this result. It goes the equation family of circles touching a given line $(y-y_1)=m(x-x_1)$ at $(x_1,y_1)$ for any value of $m$ is ...
2
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1answer
121 views

Finding a point on a circle that has a distance L (arc length) from another point

Given the coordinates of a single point on a circle and a length of an arc $L$, how do I find the coordinates of another point? Or, to put in another form: I have the radius $r$, the length of the ...
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2answers
256 views

Applications of Stein spaces in Algebraic Geometry

I want to know where are essential applications of the theory of Stein spaces in algebraic geometry. I heard Cartan's theorem A & B were used in Serre's GAGA, but are there any other applications?
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2answers
12k views

Finding an equation of a circle with a given center and a tangent line.

My math homework is finding an equation of the circle. Given that the center is at (-3,-5) and tangent to the line 12x + 5y =4. I don't know how to solve this since our professor didn't teach this to ...
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0answers
68 views

Soft: Why does the existence of a singularity cause problems for deRham cohmology?

I've heard that if a variety has a singularity then the deRham theory has "problems". What exactly are these? Im guessing there is some sort of issue with the defintion of a differential form, but ...
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2answers
55 views

Finding the equations of the lines and tangent to the circle

Find the equations of the lines through $(2,0)$ and tangent to the circle $x^2+y^2=1$. I tried to solve this and I know the right answer but I just can't solve this. The right answer: $\sqrt{3}y=x-2$ ...
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2answers
122 views

How to justify the similarity of objects in mathematics form

I have developed a system to trace the outlines of (images of) objects. Now I want to test whether two independent traces represent a common feature. Imagine two people (or machines) tracing the ...
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2answers
37 views

Prove $(|OP|)+ |PQ|)^2 > |OQ|^2$

I did all the algebra and for some reason I'm getting 0 > $y_2^2$ which is clearly wrong. Where did I mess up at?
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1answer
90 views

Equation of line passing through a point parallel to a given line

I have the point $(2,-5)$ and an equation $y-4 = 2x$ which is a straight line. I want to make another equation from the $(2,-5)$ that is parallel to $y-4 = 2x$ and you can only do this by making the ...
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2answers
140 views

Hyperbolas - Standard Form

This is probably a simple question but if $y = \frac{1}{x}$ is a hyperbola, then how does it comply with the standard form of a hyperbola?
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1answer
45 views

Given two closed curves, when is their minkowski sum differentiable?

Suppose you are given closed curves, $\gamma_1$ and $\gamma_2$, which define convex figures in the plane. If we take the minkowski sum of $\gamma_1$ and $\gamma_2$, when is the resulting curve ...
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2answers
25 views

assessing linear relationships as logarithms

I am teaching myself maths. I am not sure how to approach this problem. It is assessing linear relationships of the form $y=mx+c$ as logarithms. Here I have gotten as far as taking the gradient ($\...
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3answers
190 views

Equilateral triangle inscribed in a ellipse

"Given any point on a ellipse, is it always possible to inscribe an equilateral triangle, with a vertex coincident with that point, in the ellipse?" I thought I could use analytical geometry, but ...
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1answer
67 views

Geometric interpretation of a complex solution

A straight line in 2-D $x+y=3$ and a circle in 2-D $x^2+y^2=4$ do not have a point of intersection in the plane containing the two. But on solving these equations analytically, on gets 2 complex ...
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1answer
57 views

Equation of horizontal/vertical line and changing to $y=mx+c$ format

I've been given the equation $2x-3y=5$. I was wondering whether this is a horizontal or vertical equation and how would I rearrange this to $y=mx+c$. I know that this is a fairly basic equation but ...
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1answer
215 views

Tensor notation (practicing)

I'm praticing tensor notation, and I want to prove this way that given vectors $A,B,C,D$ then $(A \times B) \times (C \times D) = \det(A,C,D)B - \det(B,C,D)A$, where $\det$ means the triple product. I'...
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1answer
161 views

List of topics for basic calculus (1st,2nd,3rd semester)

I am an computer science student, currently studying in 2nd semester. Therefore my math courses are pretty weak. Although I "aced" them, I still feel I could use some extra basic calculus knowledge in ...
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3answers
192 views

Analytic Geometry

How does one solve: Find the equation of the circle which has it's center on the line $y= 3-x$ , and which has as tangents the lines $ 2y-x = 22, $ $ 2x+y=11 $ ?
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0answers
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Circle Tangent question

I would like to ask for assiatance on the following: Find the eqation of a circle, with a radius of$\sqrt 2$ , which also has as tangetns the lines: $ y=x+2 $ , $ y=-7x $. It is known that the ...
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2answers
220 views

Maximize the distance between a point and a bounding rectangle

There are $n$ random points in the $x-y$ plane, whose coordinates are known beforehand. We can use a minimum bounding rectangle (MBR) to bound these points. In this scenario, the MBR can be rotated, ...
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1answer
47 views

Check my solution to this trig inequality

Problem $1.88$ : Solve $$\cos x\lt \frac{\sqrt{3}}{2},\qquad x \in [0,2\pi]$$ I found the set of solutions to be $S=[0,2\pi]-\left[\dfrac{\pi}{6},\dfrac{11\pi}{6}\right]$ Is this correct? Thank you.
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2answers
581 views

Book suggestions on projective geometry

I want to be acquainted with projective geometry, so I'm asking for a reference. I need some words to explain my specific background and motivation. There are many things I learnt related to ...
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1answer
52 views

Show that the area of the shape can be written as $A=200r-r^2 (2+ \pi/2)$

A $\rm200\,m\,$ fence is to placed around a lawn of this shape. We know that $x$ in terms of $r$ : $$x=100-\dfrac{(2+\pi)r}2$$ How do I show that the area of the lawn, $A$, can be written as: $$A=...
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1answer
102 views

Tangent at a singular point

I'm looking at this question If the tangent at the point $P$ with coordinates $(h, k)$ on the curve $y^2 = 2x^3$ is perpendicular to the line $4x = 3y$, find $(h, k).$ This is how I attempted it $...
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1answer
162 views

Mirror a line over a plane

I am trying to mirror a line over a plane, but I am not sure if I am doing it right, so please tell me if something that I do is wrong. I have 2 points $A(1, 2, 1);B(-1,0,2)$ and I have to mirror the ...
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1answer
60 views

need explanation of what exactly is a directrix & focus?

((I'm not asking why do we need to know conic sections etc.) Like other similar questions.) I actually love math & currently learning conic sections in class, neither my textbook or teacher ...
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2answers
864 views

Area of triangle inscribed in a parabola

How can u prove that the area of the triangle inscribed in a parabola is twice the area of the triangle formed by the tangents at the vertices?
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1answer
274 views

Centroid of triangle formed by co-normal points

How can you prove that he centroid of a triangle formed by 3 co-normal points lies on the axis of the parabola?
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2answers
264 views

Orthogonal tangents to an ellipse [duplicate]

This is the problem I found back in the first year in the university. Suppose we have a non-degenerate (i.e. not a point and not an empty set) ellipse $E\subset \Bbb R^2$. Now define a set $D$ by a ...
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3answers
70 views

The maximum from a point outside an ellipse to a ellipse.

In the $xOy$ axes, Assume there is an ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$, and a point $A(0,t)$ ($t$ is a constant )outside the ellipse. Assume $P$ is a point in the ellipse. Find the ...
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4answers
355 views

Equal perimeter and area

Find all triangles of which perimeter and area are numerically equal. I have got solution for right angle triangles but not of others
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50 views

How find this this distance $d_{1}d_{2}=b^2$

On the plane we have two points $A(\sqrt{a^2-b^2},0),B(-\sqrt{a^2-b^2},0)$ with $a>b>0$ and the line $L$, of which the equation is given as:$$\dfrac{x\cos{\theta}}{a}+\dfrac{y\sin{\theta}}{b}=1$$...
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1answer
26 views

Is there another way to solve the value field of a parameter of an line.

Assume $P$ is a point in line $x+y=m$, where $m \in \Bbb{R}$. There are two points $A,B$ in circle $$x^2+y^2 = 10$$ such that $PA$ and $PB$ are tangent lines of the above circle. If line: $x+y=m$ has ...
2
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1answer
128 views

Finding the equation of a plane in 3-D by using point-to-point distances

Assume that we have a plane $P(a,b,c,d)$ whose equation is unknown. We know that there is a point set $N = \{n_1, n_2, ...\}$ and $\forall n_i \in N$, $n_i$ is on $P$. Also, $\forall n_i, n_j \in N$, ...
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2answers
40 views

How to compute point from {length and angle}

How to compute point from {length and angle}?
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1answer
74 views

The relation between the radiuses…

Find $\frac{R}{r}$ where $R$ is the radius of the circumscribed circle of a trapezoid and $r$ is the radius of the inscribed circle of this trapezoid. Thank you!
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1answer
310 views

Problem concerning inscribed and circumscribed circles…

Can you please help me solve this really difficult problem: Find R/r where R is the radius of the circumscribed circle of a trapezoid and r is the radius of the inscribed circle of this trapezoid. ...
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3answers
76 views

$x^2 + y^2 - y = 0$ is… a cylinder?

I've this question: Find the area of the intersection between the sphere $x^2 + y^2 + z^2 = 1$ and the cylinder $x^2 + y^2 - y = 0$. Is this second equation even a closed shape? If one were to ...
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2answers
2k views

Perpendicular form of the straight line equation.

There are 5 to 6 standard forms of the straight line equation. for example slope intercept form, two intercept form, point slope form and perpendicular form. I have clear visualization of all forms ...
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1answer
63 views

Lattice Points on a straight line.

To find: The number of lattice points in the 1st quadrant, lying on straight line: 3x 5y = 283. -I tried this question a lot many times. The long substitution method becomes tedious. Can u please ...
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37 views

sides of two triangles which have different areas

consider 2 triangles like $\bigtriangleup ABC \quad and \quad \bigtriangleup \acute{A}\acute{B}\acute{C}$, which $S_{\bigtriangleup \acute{A}\acute{B}\acute{C}} \leq S_{\bigtriangleup ABC}$.(S stands ...
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224 views

How to show that a regular pentagon can't have all coordinates rational

This is pretty straightforward if we're allowed to use trigonometry, so I guess my question is Are there any nice (trigonometry-less) proofs of the fact that a regular pentagon in the plane must ...
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2answers
44 views

Find perpendicular vectors in subspace of $V_{3}$

Find all vectors of $V_{3}$ which are perpendicular to the vector $(7,0,-7)$ and belong to the subspace $L((0,-1,4), (6,-3,0)$. As a note, this is an extra question of a long exercise, the vectors ...
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63 views

Prove $\sin^2 A = \sin^2 B \sin^2 C - 2\sin B \sin C \cos A$

I am asking for help with this proof: Given $\triangle ABC$. Prove that $\sin^2 A = \sin^2 B+ \sin^2 C - 2\sin B \sin C \cos A$
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2answers
77 views

$\overline{x} \times \overline{a} = \overline{b}$ has a solution when $ \langle\overline{a},\overline{b} \rangle =0$

I'm trying to solve this exercise: Let $\overline{a} \neq \overline{0}$, $\overline{b}$ be two vectors of the Euclidean vector space $V_{3}$. Prove the equation $\overline{x} \times \overline{a} = ...
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2answers
69 views

How can I find the coordinates of a point which is the reflection of a point about a line in 3D

I am currently working on a project on Matlab and I need to find the coordinates of a point which is reflected about a line. I know how to do it in 2D but in 3D things are getting ugly. So, we have a ...
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1answer
42 views

Scalar times Point + Scalar times Point?

Let $P$, $Q$ be a pair of points in the Euclidean plane and let $t_1$, $t_2$ be a pair of scalars. My textbook says that the following operations are nonsense: $$P + Q\\ t_1 \cdot P$$ However $t_1 \...