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

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1answer
16 views

What is the basic idea of homogenisation of an equation?

I do get that when you are homogenising it makes it in an equation of pair of straight lines passing through origin but what is its actual point and its applications?
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1answer
34 views

Lines joining origin to points of intersection of two conics

If the lines joining origin and point of intersection of curves $$ax^2+2hxy+by^2+2gx=0$$ and $$a_1x^2+2h_1xy+b_1y^2+2g_1x=0$$ are mutually perpendicular, then prove that $$g(a_1+b_1)=g_1(a+b)$$ How ...
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1answer
15 views

Coordinate Geometry of circles; Radical Axis question

If one of the diameters of the circle $x^2+y^2-2x-6y+6=0$ is a chord to the circle with center at $(2, 1)$, then the radius of the second circle is? Apparently the solution is $3$, with the ...
2
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1answer
48 views

Help with simple rotation on an x,y plane

I'm a programmer, with too little background in mathematics, and I am currently faced with the challenge of rotating an object on a 2 axis plane. Something that is hopefully quite easy for you guys. ...
2
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2answers
35 views

Calculus III: Find the points of the curve…

I have to find the points of the curve $$r\left( t \right) =\left( t,{ t }^{ 2 },{ t }^{ 3 } \right) $$ where the osculating plane passes through the point $\left( 2,-\frac { 1 }{ 3 } ,-6 \right)$.
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1answer
12 views

Coordinates rotation and function change

In the Cartesian coordinates $(x,y)$, I have a vector function $\bar{f}(x)=\hat{x}A\cos(yk)$, where $A$ and $k$ are constants. I make now a 45 degrees rotation (in the same plane) to the new set of ...
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1answer
26 views

Three planar vectors $x,y,z$ such that $x$ is orthogonal to $y + z$ and $z$

Let $x$ be a non-zero vector, orthogonal to vectors $y + z$ and $z$, with $x, y, z \in \mathbb R^2$. Prove that $y$, $y - z$ and $z - y$ are orthogonal to $x$ and parallel to $z$. To prove they ...
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0answers
40 views

intersection of a line and plane on a 3-sphere

Suppose I have two 4D points, $\mathbf{a}=(a_1,a_2,a_3,a_4)$ and $\mathbf{b}=(b_1,b_2,b_3,b_4)$, that both lie on a unit 3-sphere (i.e. unit distance from origin). In addition, I have a 2-D plane that ...
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3answers
111 views
+50

Constructing a family of distinct curves with identical area and perimeter

Two recent questions were posed by Arjuba [1] [2] asking for counter-examples regarding whether two different figures could have the same perimeter and area. Responders quickly raised a number of such ...
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1answer
12 views

Equation of a line with a positive gradient [closed]

Two straight lines passing through the point (0,2) are tangent to the graph of the function y=1-x^2. Find the equation of the line with a positive gradient.
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2answers
38 views

Determine the matrix of the reflections in the fol­lowing plane in $\Bbb R^3$. [closed]

$2x_1-2x_2-x_3= 0$ How would I go about approaching this problem?
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2answers
51 views

Parametric formula for figure 8 mobius strip

I'm making 3D prints with Mathematica, and am interested in a parametric formula for a mobius strip that is in the form of a figure 8, rather than simply a circle with a twist in it. Can someone help ...
0
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1answer
23 views

Equation for the length of a chord parallel to either the minor or major axis in an ellipse

I am looking for a way to compute the length of any chord parallel to the minor (or major) axis of an ellipse. In all cases I know the lengths of both axes, and the distance between the chord and axis ...
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0answers
29 views

Need a solution to find the locus of an equation. [closed]

Find the equation of the locus of a point which moves so that its distance from $(a,0)$ is equal to its distance from the $y$-axis. The answer is $$y^2 - 2ax + a^2 =0$$ Please can someone find the ...
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0answers
4 views

Find gap coordinates to connectable reference Rectangle

Sorry for my english. This is my first question. I need to find gap coordinates of items to reference object programmaticly. Item positions can be changed. How can i do this with an efficent way? ...
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1answer
23 views

Line not intersecting circle, maximum value of expression involving radius

If line $y+x=2$ do not intersect any member of circles $x^2 + y^2 -ax = 0$ at two distinct points where a is parameter, then maximum value of $|a + 4|$. My try: Since the line does not intersect ...
0
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1answer
27 views

Straight Lines co ordinate geometry

At what angle with the line x+y=4, a line through (1,2) be drawn so that the distance between the point of intersection of the lines and the point (1,2) is 6/(root 3)?
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3answers
64 views

getting the slopes of the sides of an equilateral triangle given 2 points

I want to get the slopes of an equilateral triangle given the 2 vertices. Let's say they are (0, 0) and (5, 5). Graphing this would give 2 triangles forming a diamond. I tried to use distance formula ...
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2answers
46 views

How to parameterize the maximum of a function?

$f(x) := -4(\frac{1}{2}-x)^2+1$ Is it possible to construct a parameterized version of $f$, say $f_a$, which fulfills: $f_a(0) = f_a(1) = 0$ $f_a(a) = 1$ $f_a'(x) = 0 \Leftrightarrow x=a$ $f_a''(a) ...
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3answers
67 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
204 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
25 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
38 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
16 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 ...
0
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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
57 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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4answers
52 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
27 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
24 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 ...
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3answers
479 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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0answers
20 views

Find the properties of a tilted, off-center ellipse from its general equation

I have an ellipse with this general formula: $$0.00228797 x^2+0.00138781 x y-0.281261 x+0.00209387 y^2-0.832702 y-1.43328=0$$ I want to find out about its center (h,k), axis half-lengths (a,b) and ...
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1answer
76 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
39 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 ...
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1answer
31 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
31 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 ...
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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
26 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
53 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
40 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 ...
1
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2answers
74 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
14 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 ...