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

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Graphs interpretation question

Suppose we have a prarbola $y^2 = 2px$ ....this is in fact $y = \sqrt{2px}$, so we plot it like a square root function, so it has no applied values for less than zero. However I saw in my textbook ...
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1answer
14 views

About a pair of vectors and the value of its sum norm

Knowing that |u|=11, |v|=23 and |u-v|=30 how can i calculate |u+v| (where || denotes the norm of a vector)?
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1answer
18 views

Ellipse cutting orthogonally

If the curves $ax^2+by^2=1$ and $a'x^2+b'y^2=1$ cut orthogonally, then : A)$\displaystyle \frac{1}{b}+\frac{1}{b'}=\frac{1}{a}+\frac{1}{a'}$ B)$\displaystyle ...
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3answers
59 views

Understanding vector projection

I'm learning about vector projection. I understand how to perform it, but I still can't understand what it actually means and what it gives me. Here is a common definition: Vector projection of a ...
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2answers
100 views

Find an unknown coefficient in a line equation…

So, I have to find the unknown coefficient in this line: $$x+y+C=0$$ so that it is a tangent to this circle: $$x^2+y^2-5x-7y+6=0$$ I've transformed the circle equation to this form: ...
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Movements in the complex projective plane.

My textbook denotes movements in the Euclidean plane by $P(a,b,\alpha):\mathbb R^2\to\mathbb R^2$. Each movement depends on three numbers $a,b,\alpha\in\mathbb R$ and is given by $(1)$ $$(1)\quad ...
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1answer
75 views

How do i find the radius and the center of these circles

Please help me with my math Radius Homework Help is really appreciated! PS: Pls Don't be bothered by my erasure on my sheet, those 4 question are unanswered. Instructors' formula is x^2 + y^2 = r^2 ...
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2answers
67 views

How do i Solve the Radius of the circle?

Hello I was wondering about this kind of problem I'm having. Here it is: $$ x^2 + y^2 = 49 $$ Formula given by our instructor is: $x^2 + y^2 = r^2$.
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3answers
189 views

Distance of a point to a line

Find the distance from the point $S(2,2,1)$ to the line $x=2+t,y=2+t,z=2+t$. How can I find the distance of a point in $3D$ to a line?
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1answer
138 views

Analytic geometry - rotation + translation

In $K=O\vec{e_1}\vec{e_2}\vec{e_3}$ I have to find the analytical representation of the screw motion( rotation + translation) $\psi$ with a rotation axis $g$ given by the points $A(5,-4,3)$ and ...
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53 views

Center of circumscribed circle of a triangle

I've been given the parametric equations of the height, median and inner angle bisector through the point $A$ of a triangle $\triangle{ABC}$: $$ h: \begin{cases}x = 2 - s \\ y = 1 \\ z = -3 + 2s ...
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1answer
50 views

Why does the area of an area element increase by $1/cos(\theta)$ after tilting it?

While reading this chapter of the Feynman Lectures I came across a statement I didn't know how to prove. He mentions below Eq. 4.30 that when you take a surface and tilt it by some angle $\theta$, ...
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1answer
90 views

Rational Point in circle

How many rational point(s) (a point (a, b) is called rational, if a and b are rational numbers) can exist on the circumference of a circle having centre (pie, e)
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2answers
60 views

Closest Point on a Sphere to Another Point

Given a sphere $S(c,r)$, $c$ being the center point $(x,y,z)$ and $r$ being the radius, there is a point $p(x', y', z')$ which is either inside or outside $S$. I want to find the point $q$ such that ...
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1answer
153 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: ...
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0answers
44 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? ...
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1answer
80 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 ...
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1answer
34 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 ...
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4answers
3k 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 ...
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2answers
25 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{ ...
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4answers
94 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.
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1answer
49 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!
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1answer
126 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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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 ...
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2answers
66 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 ...
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2answers
26 views

How to create perpendicular bisector

Say we have an 0XY coordinates plane. We have coordinates of points A(xa, ya), B(xb, yb) ...
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41 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 $M$ ...
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3answers
132 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 ...
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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 ...
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1answer
68 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 ...
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1answer
129 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
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1answer
45 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 ...
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1answer
58 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} = ...
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2answers
47 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 ...
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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 ...
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1answer
45 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 ...
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0answers
25 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 ...
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61 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?
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44 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 ...
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57 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 ...
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1answer
26 views

Vectors triangle problem.

Let D be the midpoint of the side BC of the triangle ABC Verify that: $$\vec{AD}=\tfrac12\big(\vec{AB}+\vec{AC}\big)$$
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171 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 ...
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80 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 ...
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1answer
67 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 ...
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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, ...
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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, ...
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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 ...
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159 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 ...
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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 ...
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1answer
53 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 ...