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

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3answers
69 views

Curiosity about Kronecker's Delta?

My professor gave this subject in the class (analytic geometry) and I thought it was very complicated, then I just decided to open Wikipedia entry on Kronecker's Delta and discovered it is quite ...
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2answers
103 views

Omitting $i$ in calculations

Is it possible in various calculations related to the complex plane which also include analytic geometry , calculating distances etc, to omit $i$ and treat the imaginary axis as simply the cartesian ...
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3answers
210 views

Calculate the distance from a point to a line

Por favor, alguém me ajude com essa questão de Geometria: Please, can someone help me with this geometry question? Given the point $A(3,4,-2)$ and the line $$r:\left\{\begin{array}{l} x = 1 + t ...
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1answer
28 views

transform line and point in 3d and 2d space [closed]

I have a line which is described with two point and I know (x0,y0,z0) and (x1,y1,z1). After that I transform it to 2d space dividing with -z0 and -z1 values. Problem is that if I know (a,b) how can ...
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1answer
39 views

Central angle of an ellipse

If I have an ellipse centered at the origin and know the length of $a$ and $b$ and was given the length of an arc, how can I find the angle that is between the two radius from the center of the ...
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1answer
19 views

Preserving incidence relation proof

How can one prove via analytic method that projective map preserves incidence relation?
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0answers
23 views

normalized mean curvature flow with convex initial hypersurface has finite velocity

I can't understand the prove in [Xi-Ping Zhu] Lectures on mean curvature flows. The statement as follow. Lemma 3.5 (page 32) There exists a positive constant $C$ such that ...
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1answer
26 views

Line parallel to plane

See if the line e is parallel with to the plane $α$. If not, find the intersecting point. $$\begin{align} α: & \quad \quad x-3y+z+1=0 \\ e: & \quad \{x+y-z=3, 2x-y-4z=3\} \end{align}$$
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1answer
69 views

Equation of parabola, tangent at vertex [closed]

Two tangents on a parabola are $x-y=0$ and $x+y=0$. If $(2,3)$ is the focus of the parabola, then find the equation of tangent at the vertex. Thanks. My thoughts: Can't figure out anything :(
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1answer
19 views

Determine Center Point based on 2 separate elipses

First timer here. I've been digging back into my good old maths days but am extremely rusty (beyond belief). I got a really tricky question that i want to determine formula for so that my mate can ...
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votes
4answers
53 views

Find the line through $(-1,4)$ for which the distance to $(6,3)$ is 5

This is the question: Find the line through $(-1,4)$ for which the distance to $(6,3)$ is $5$ The answer is: $y-4=-4/3(x+1)$ and $y-4=3/4(x+1)$ I do not know how to get this answer. ...
2
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1answer
28 views

Definition of (hyper)planes

I know the definition of a plane to be: $(r-r_0)\cdot n = 0$ where $n$ is the vector perpendicular to the plane, $r$ the vector to a given point and $r_0$ the vectors to the points which constitute ...
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4answers
25 views

Showing that a circle is “tangent” to the $x$-axis if and only if $\left|k\right| = r$.

The problem is this: to show that a circle of radius $r$ and center $(h, k)$ intersects the $x$-axis at exactly one point if and only if $\left|k\right| = r$. Using geometrical intuition, this ...
7
votes
3answers
489 views

Can asymptotes be curved?

When I was first introduced to the idea of an asymptote, I was taught about horizontal asymptotes (of form $y=a$) and vertical ones ( of form $x=b$). I was then shown oblique asymptotes-- slanted ...
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1answer
80 views

How can I convert the following parametric equation to cartesian equation?

\begin{align} x&=\left(1 + \frac{1}{\,\sqrt{\,2t^{2} - 4t + 4\,}\,}\right)t\ -\ 2 \\[3mm] y&=\left(1 - \frac{1}{\,\sqrt{\,2t^{2} - 4t + 4\,}\,}\right)t\ +\ \frac{2}{\,\sqrt{\,2t^{2} - 4t + ...
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0answers
22 views

How can I find the volume of this prism and points B, C, D and F?

In the triangular prism, A = (0, -1, 1), E = (0, -3, 0). C and D belong to line s: x - 1 = y = y - z. How can I find the prism's volume and the coordinates of B, C, D and F points?
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3answers
44 views

Find a specific vector equation of a line that divides a angle in half.

I've been studying a little geometry on my own, and I just recently stumbled on this problem, that I'm unable to answer: Given the points A=(2,-1), B=(5,4) and C=(-7,8), find a vector equation of a ...
1
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1answer
33 views

Doubt on locus of a median point

I'm learning about geometric locus and ain't had an good time, I'm struggling with this problem: By the way, any study resource on geometric locus is welcome! Given an segment $AB$ formed by points ...
1
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1answer
32 views

finding a point of intersection

I need to find a point on the $y-axis$ so the tangents from that point to circles: $(x-6)^2+(y-3)^2=16$, $(x-4)^2+(y-6)^2=5$ are equal in length. I tried to use $(x-a)(x_1-a)+(y-b)(y_1-b)$ but it ...
0
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1answer
23 views

The number of points in the rectangle which lie on the curve $y^2=x+\sin x$ and at which the tangent to the curve is parallel to the $X-$axis

The number of points in the rectangle : $\{(x,y)|-10\le x\le10$ and $-3\le y\le3\}$ which lie on the curve $y^2=x+\sin x$ and at which the tangent to the curve is parallel to the $X-$axis, is A) $0$ ...
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3answers
24 views

Analytic geometry - Mutual tangent for circle and ellipse

The problem I'm trying to solve is : Given a circle of equation $x^2+y^2=4$ ,an ellipse of equation $2x^2+5y^2=10$ and their mutual tangent whose equation is $y=kx+n$, determine $k^2+n^2$. I would ...
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1answer
31 views

Equation of pair of reflected straight lines given the equation of pair of incident straight lines

If $ax^2 + 2bxy + by^2 = 0$ represents a pair of lines, then find the combined equation of lines that can be obtained by reflecting these lines about the x-axis. I know that this can be done by ...
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2answers
56 views

Locus of vertex of triangle moving inside circle

A right triangle with sides $3,4$ and $5$ lies inside the circle $2x^2+2y^2=25$. The triangle is moved inside the circle in such a way that its hypotenuse always forms a chord of the circle. The locus ...
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1answer
38 views

What is the approach required for questions in which you least expect that the graphs meet?

Find the no. of solutions of x in these two equations: (A)$2^x=x^2+1$ (B)$e^x=2x^2$ Both are of the same type, that is, the answer is the least you can expect. (When you plot it on a grapher, you ...
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1answer
32 views

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
13 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
17 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
43 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
79 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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0answers
15 views

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
68 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
63 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
174 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
105 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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0answers
44 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
46 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
49 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
43 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
53 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
27 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
73 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
32 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
571 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
22 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
66 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
24 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
53 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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0answers
23 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
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2answers
56 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
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) ...