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

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3
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1answer
85 views

Given an ellipse's center, focus and point, find its equation.

Given an ellipse's center is $(2,1)$, focus is (2,4) and point is (3,-3), we have Plug in center: $\frac{(x-2)^2}{a^2}+\frac{(y-1)^2}{b^2} = 1$ Use focus: $4^2=a^2-b^2$ $16=a^2-b^2$ Use point: ...
1
vote
0answers
36 views

Given equation of parabola, find vertex and directrix

Given that $x^2-bx+17-ay=0$ has vertex $(3,2)$, find the directrix and focus. My attempt is to make it into the form $(x-h)^2=4a(y-k)$ which has focus $(h,k+a)$ and directrix $y=k-a$. Is this right? ...
8
votes
1answer
75 views

Zariski vs analytic cohomology of $\mathcal O_X^\times$

Let $X/\mathbf C$ be a smooth proper variety. Is it true that $H^1(X, \mathcal O_X^\times) = H^1(X^{an}, \mathcal O_{X^{an}}^\times)?$ GAGA doesn't apply, because $\mathcal O_X^\times$ is not coherent ...
2
votes
1answer
33 views

$\frac{|| \overline{AM}||}{|| \overline{AB}||}=\frac{|| \overline{AN}||}{|| \overline{AC}||}=\frac{|| \overline{MN}||}{|| \overline{BC}||}$

$\Delta ABC$ is a triangle, $M$ is a point in the segment $\overrightarrow{AB}$ and $N$ is a point in the segment $\overrightarrow{AC}$, such that $\overrightarrow{MN}$ is parallel to ...
0
votes
4answers
2k views

Finding an equation of circle which passes through three points

How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$ using the formula $(x-q)^2 + (y-p)^2 = r^2$. I know i need to use that formula but have no idea how to ...
0
votes
2answers
24 views

Finding the equations of surfaces of revolution

I have the following question: $$\text{Sketch and find the equations of the surfaces formed by}$$ $$\text{i) }x^2 - y^2 + 1 = 0 \text{ about the y-axis}$$ $$\text{ii) }x^2 - 2y^2 + 2a^2 = 0 \text{ ...
0
votes
4answers
79 views

Calculate the angles of a isosceles triangle

In the triangle below, is there a way to calculate the $x$ and $y$? To be more specific, $b = 12.8\rm\,cm\ $ and $h = 10\rm\,cm$, hence $a = 11.87\rm\,cm$. I don't know what to do from here.
2
votes
1answer
39 views

proof for euler-rodrigues formula - matrix form

I need for a matrix representation. Exactly I want to know how to get the Euler-Rodrigues formula in a matrix form like here. Thanks!
1
vote
1answer
80 views

Calculating the adjustment translation to be applied after rotating and scaling so that operations pivot about a given point.

I have a matrix for transforming an image into a target frame. The matrix is a function of a scale, $s$ rotation angle, $\theta$, and a translation that is applied after rotating, $tx, ty$. The ...
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votes
0answers
25 views

Functional Relationship Question on Analytic Geometry

I am solving some problems on analytic geometry. I have a set of points $\{P_1,P_2,P_3,...,P_k\}$ from wich $P_1,P_2$ are known. The rest have coordinates $P_n\big(x_n,y_n\big)$ and for any value of ...
1
vote
2answers
59 views

Solving a geometric question without trigonometric tools.

$AB$ is a diameter in a circle. from point $C$ outside the circle passing $2$ intersects to the circle at points $A$ and $B$. $AC$ cuts the circle at point $F$ and $BC$ cuts the circle at point ...
0
votes
2answers
24 views

How to create perpendicular bisector

Say we have an 0XY coordinates plane. We have coordinates of points A(xa, ya), B(xb, yb) ...
0
votes
0answers
39 views

confusion about real analytic manifold

If $M$ is a real analytic manifold, let $X$ be a complex neighborhood of $M$ and $i: M \hookrightarrow X$ be the canonical inclusion. Let $\mathcal{O}_M$ be the sheaf of real analytic functions on $M$ ...
1
vote
3answers
113 views

Moments at which moving points on a circle coincide

Points A $(0,1)$ and B $(1,0)$ start moving along the circumference of a unit circle with center $(0,0)$ in the same, positive (that is, counterclockwise) direction. Every minute, points A and B ...
0
votes
0answers
24 views

n-gon Inequality Theorem Converse

In the plane, if we have an n-gon with side lengths $v_1$, $\ldots$,$v_n$, these lengths satisfy the "planar $n$-gon inequalities," ie. the length of each side is less than the sum of the lengths of ...
1
vote
1answer
60 views

Finding the intersection of a circle and a line

The text says: On a single set of coordinate axes, sketch the line $x+16 = 7y$ and circle $x^2+y^2-4x+2y=20$ and find their points of intersection. Hint: eliminate x algebraically and solve the ...
3
votes
1answer
109 views

How to determine if arbitrary point lies inside or outside a conic

Given the general equation of a conic $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $, is there a way to determine if an arbitrary point $(x_1,y_1)$ lies inside or outside of the conic (ex. parabola or ...
2
votes
1answer
44 views

Algebraic Compact manifold originates from a proper scheme?

If $M$ is a compact complex manifold, which is the analytification of some scheme $X$ of finite type over $\operatorname{Spec}(\mathbb{C})$, then must $X$ be proper over ...
3
votes
1answer
51 views

Area enclosed between half lines in polar space

I don't know if the anwser to my question is obvious because I cannot find any explanation anywhere on google. Question The blue region $R$ is bounded by the curve C with equation $r^{2} = ...
0
votes
2answers
45 views

Length of a line in an isosceles triangle. (mind boggling )

In an isosceles triangle ABC, side AB and AC are equal in length. There exists a point D on the side AB. The angle BAC is theeta . The side AD is two units smaller than AC .What is the generalized ...
26
votes
3answers
1k views

Fascinating Lampshade Geometry

Today, I encountered a rather fascinating problem in a waiting room, which is embodied in the image below. Notice how the light is being cast on the wall? There is a curve that defines the ...
0
votes
1answer
40 views

To find an intersection point between two planes with only the direction vector

Find the intersection between two planes $x−3y−2z = 2$ and $2x+y+3z = 1$ Solution: $(1)$$\quad n_1 \times n_2 =\langle −7,7,7\rangle =7 \langle −1,1,1\rangle$. $(2)$ To find one intersection ...
2
votes
0answers
20 views

finite morphism (algebraic) vs finite morphism (analytic)

Let $X$ and $Y$ be two algebraic varieties (reduced schemes of finite type) over $\mathbb{C}$. Let $f : X \to Y$ be a morphism of schemes. Let $X^{an}$, $Y^{an}$ and $f^{an}$ the corresponding ...
0
votes
1answer
58 views

How to find the midpoint given the linear equation

Given the linear equation: $$y = 7247.5188 -2395.0376x$$ how do I find the midpoint of this line?
2
votes
0answers
41 views

Locus of centre of circle in Lambert theorem

A beautiful theorem, when three tangents to a parabola form a triangle,the focus of the parabola lies on the circumcircle of the triangle. But what is the locus of the centre of the circumcircle of ...
3
votes
0answers
53 views

Is there an algebraic description of the ring of analytic functions on the real projective line?

Apologies for the long question. Let $X=\mathbf P^1(\mathbf R) \subseteq \mathbf P^1(\mathbf C)$ be the real projective line. Let $\mathcal O_X$ be the sheaf of real-analytic complex-valued functions ...
2
votes
1answer
26 views

Vectors triangle problem.

Let D be the midpoint of the side BC of the triangle ABC Verify that: $$\vec{AD}=\tfrac12\big(\vec{AB}+\vec{AC}\big)$$
1
vote
2answers
102 views

how to calculate the angle between the tangents of the curve?

$y=(-3/2)x$ and $y=(-2/5)x$ intersect the curve $$3x^2+4xy+5y^2-4=0$$ at points $P$ and $Q$ .find the angle between tangents drawn to curve at $P$ and $Q$ .I know a very long method of finding ...
4
votes
0answers
72 views

Tangent developable of helix.

Let $T$ be union of tangent lines to helix $C=(\cos x, \sin x,x)$. 1) I want to prove that $T - C$ is a smooth manifold and find equation for $T$. 2) I want to find how many times a line can ...
1
vote
1answer
56 views

Find all points with whole- number's coordinates inside the area of polygon

I've got the polygon with n angles. I know the coordinates of its apexes (their coordinates are integers), but I don't know the total area of that polygon. Is there any way to count how many points ...
0
votes
0answers
39 views

Computer program for visualizing multivariable Calculus topics

I am an undergraduate studying multivariable Calculus. However I have difficulty visualizing concepts. In single variable calculus I can visualize stuff, for example when one talks about derivatives, ...
1
vote
0answers
44 views

building a polytop from polytop and finding its volume

Let $P$ be a symmetric polytope with $M$ vertices. Suppose we subdivide this polytope into $M$ equal parts $A_i, i=1, \ldots, M$ such that each part $A_i$ correspond to one vertex, $v_i, i=1, \ldots, ...
0
votes
1answer
18 views

About a/the definition of plane.

Let $P$ be a point in 3-space and consider a located vector $ \overrightarrow {0N}$. We define the plane passing through $P$ perpendicular to $ \overrightarrow {0N}$ to be the collection of all ...
0
votes
1answer
70 views

Shortest distance between parallel line and plane

I've been doing questions regarding the shortest distance between lines/planes and points , and I've come across a question asking to find the shortest distance between a line and a plane which are ...
3
votes
0answers
35 views

Applications of the quartic curve $x^2y^2-1=0$?

The quartic curve $x^2y^2-1=0$ is equivalent to the union of the hyperbolas $xy-1=0$ and $xy+1=0$, i.e., it's a rectangular hypobola superimposed with a copy of itself rotated by 90 degrees. Does this ...
1
vote
1answer
50 views

Calculating tangent on ellipse

I want to calculate the slope of the tangent at one point of an ellipse whose centre is shifted towards the coordinates $(x_c;y_c)$ and also rotated by an angle $\alpha$ around its centre. Now, I have ...
1
vote
4answers
101 views

A triangle has to find its third side.

Problem: (Euclid had a triangle in mind - I am including this line so that future googles come across this question) The triangles longest side is $20$ and another side is $10$. Its area is $80$. ...
1
vote
1answer
48 views

Crazy rectangles, semi-circles, and circles!

Problem is to find the ratio of the area of the circle to that of the semi-circle. Note that points $F$ and $E$ weren't given in the original diagram, and that the circle at the top-right ...
0
votes
1answer
21 views

Proving congruency of triangles

Question: Given $AB$ is diameter, $C$ and $D$ lie on circumference, $AB = 15cm$, $AC = 12cm$, $BD = 9cm$, find area of quadrilateral ABCD. Note that the points $O$ and $Q$ were not in the ...
1
vote
2answers
46 views

What is a homographic solution in three body problem?

I came across Saari's homographic conjecture in Three Body problem. I need more information on what exactly is a homographic solution and how is it different from a homothetic solution?
0
votes
1answer
24 views

Locus of the centre of a circle $\Gamma$

Let $\Gamma_1,\Gamma_2$ be two circles centred at the points $(a,0),(b,0);0<a<b$ and having radii $a,b$ respectively.Let $\Gamma$ be the circle touching $\Gamma_1$ externally and $\Gamma_2$ ...
0
votes
1answer
19 views

How to prove that given a line L prove that all points of a fixed distance k form two lines parallel to L

How can I prove start this?I know intuitively since they never meet they are parallel, but I don't think that is a direct proof.
2
votes
0answers
25 views

Is there a Focal Point/Area/Line of a Parabola for not perpendicular Lines

I'm not sure if this is mathematical enough for this forum, since it's my first post, but please don't be too harsh! So my question is: If the incoming lines of a Parabola come in perpendicular to ...
1
vote
1answer
62 views

Conic Sections and Complex numbers

If $\omega$ is a complex number such that |$\omega$| does not equal 1, then the complex number $$z = \omega + \frac{1}{\omega}$$ describes a conic. The distance between the foci of the conic described ...
1
vote
1answer
46 views

Geometric Intuition Of Partial Derivative in 2D space

$$f(x,y)=3x^2+5y^2-2x+3y+7=0\\f'x=6x-2=0\\x=\frac {1}{3}\\f'y=10y+3=0\\ y=\frac {-3}{10} \\ O(\frac {1}{3},\frac {-3}{10})$$ As you see, we've used the partial derivative to find the center point of ...
0
votes
1answer
26 views

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?
0
votes
1answer
48 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, ...
0
votes
1answer
17 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 ...
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vote
0answers
54 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 ...
1
vote
1answer
57 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 ...