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

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2answers
23 views

Is it possible to find equation for ellipse when focus and two points are known?

Is it possible to find equation for an ellipse when we know two points and one focus in 2d cartesian coordinate system? We can also make these assumptions about these two given points depending on ...
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0answers
37 views

How can I find the common axis of 2 cones in space that have the same base radius but different heights?

How do I find the 3D vector describing the axis of 2 overlapping cones, like this: If I have only the following information: Coordinates of the common tip Coordinates of a point on the yellow ...
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0answers
14 views

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$. [duplicate]

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$. Here $P(1,-1,7)$ and $Q(7,5,1)$. I have tried to find $r(t)$ by using the formula $r(t)=p+t(p-q)$ but ...
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2answers
16 views

Why $(h,k)$ in equation $y= a(x-h)^2 +k$ is the vertex of a parabola?

As in the title , I know how to convert normal explicit equation to a vertex form equation by completing the square . But what is the reasoning behind why $(h,k)$ must be the vertex , but not other ...
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2answers
36 views

What does 'forms a right-handed set' mean?

In a question I am reading, the following question appears. What if $\vec{A},\vec{B}$, and $\vec{C}$ are mutually perpendicular and form a right-handed set? What exactly does "form a ...
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2answers
26 views

Hyperbola question

the graph $ y^2=16x $ is a hyperbola; it can be rewritten as $ y= \pm 4\sqrt{x}$ when I draw it down however It is clearly not a function..question is whether it has to be one in order to perform ...
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1answer
22 views

Simple Analytic Geometry Problem [closed]

Could someone explain to me the solution of this problem? Find the equation of a circle passing through (3,7) and tangent to the lines x-3y +8=0 and y=3x
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1answer
14 views

What is the vector equation of the line through the head of $v_0$ and parallel to $v_p$?

$v_0$ and $v_p$ are vectors. Let $v_0, v_1$ and $v$ be vectors, all emanating from $(0, 0, 0)$. Suppose the line $l$ is passing through their heads. Let $v_p$ be on the line $l$ such that $v_1 = v_0 ...
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0answers
14 views

About parametric equation of a line in $3$-space

$a.$ Given coordinates $(x, y, z )$ with origin $(0,0,0)$, parameterize the line through the points $(4,5,6)$ and $(1,2,3).$ $b.$ Take components of your answer to Part $(a)$ to give three ...
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0answers
28 views

Endless repeating tiles

Regarding to this question ( "Hall of mirrors" OR "wraped plane" - Problem ) there's still an open point: how can i determine if the sum of all vectors pointing from p1 to p2 is ...
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2answers
15 views

finding the ratio which divides the segment

Find the ratio in which the point (2,-1) divides the segment from (6,1) to (0,-2). Find the coordinates of the point that divides the segment from (0,-1) to (6,3) I the ratio 2:5 . can somebody ...
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2answers
21 views

How to find the equation of $f(x^3)$ given the tangent in $f(x)$? [closed]

The exercise says: The tangent of $f(x)$ in $x=1$ is $y=2x-1$. Find the tangent of $y=f(x^3)$ in $x=1$.
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1answer
19 views

finding a soild from five planes

Given five planes: $\pi_1=2x+5y+z-2=0$ $\pi_2=x+y-z-1=0$ $\pi_3=x+4y+2z-4=0$ $\pi_4=3x-y+4z-3=0$ $\pi_5=-6x+2y-8z+k=0$. How can i find the solid shape that is formed by those planes? I tried to ...
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2answers
47 views

About the sum of sines of two angles

Suppose that $0\le \alpha\le \pi/2$ and $0\le \beta\le \pi/2$ such that $\alpha+\beta\ge \pi/2$. Can we prove that $\sin(\alpha)+\sin(\beta)\ge 1$?
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1answer
43 views

Why ternary diagrams work

I am trying to understand why ternary diagrams work. In order that the altitude criterion be valid, if I correctly understand, given equilateral triangle $ABC$, whose vertices I name as the three ...
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3answers
40 views

Describe geometry of complex plane.

Let $a \in \Bbb R$ and $c>0$ to be fixed. Describe the set of points $z$ such that $|z-a|-|z+a|=2c$ for every possible choice of $a$ and $c$. Then let $a$ be a complex number using the rotation ...
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2answers
26 views

Equation of tangent from a point outside it

Is there any general method of finding the equation of the tangent of a function $f(x)$ from a point $(a,b)$? $\hspace{1 mm}$ Then how do you find the angle between two tangents from $(0,0)$ to a ...
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4answers
230 views

Is there a name for the curve $t \mapsto (t,t^2,t^3)$?

Is there a name for the curve given by the parametrization $\{(t,t^2,t^3); t\in\mathbb R\}$? Here is a plot from WA. An another plot for $t$ from $0$ to $1$. This curve is an example of a ...
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1answer
20 views

Point coordinates at a fixed distance from a vector

I would like to solve the following generic problem by using vector notation that I will use it to improve my algorithm. I have a vector P1P2 that points P1 and P2 are known. Furthermore, an ...
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1answer
33 views

Why does $(a-2b)\times (3a+2b) = a\times (3a+2b) - 2b \times(3a+2b)$?

Let $\textbf{a},\textbf{b}\in\mathbb{R}^3$ be such that $|\textbf{a}| = |\textbf{b}|$ and the angle between them is $45º$. We had a test where we should find the answer of ...
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2answers
54 views

Change of basis matrix for polynomials?

I've understood what a change of basis matrix is, and how it's structured. So a change of basis matrix from $B$ to $C$ is the matrix $M$ such that: $${\begin{bmatrix} &\\ \\ \\\end{bmatrix}}_B ...
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6answers
86 views

Intersection of a sphere and a plane

How can I find the intersection between the sphere $x^2+y^2+z^2=1$ and the plane $x+y+z=1?$ Context This is related to a computation of surface integral using Stokes' theorem, Calculate the surface ...
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3answers
77 views

Coordinate of the excentre of a triangle

I am just wondering that how the coordinate of the excentre comes out if we know the coordinates of vertices of the triangle.
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2answers
42 views

Finding midpoint of rectangle in 3D vectors

If given the points (-10,-2,0), (-10,2,0), (-12,0,2) and (-12,0,-2), how do I find the midpoint?
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1answer
28 views

Find the map of the closed ball $B(0,1)$ of the following continuous function $f(x,y,z)=(\frac x3,\frac y2-1,\frac z9+1)$ and $f^{-1}(0)$.

Find the map of the closed ball $B(0,1)$ of the following continuous function $$f(x,y,z)=\left(\frac x3,\frac y2-1,\frac z9+1\right)$$ and $f^{-1}(0)$. $f^{-1}$ seems quite simple, I got ...
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1answer
25 views

Focal length of an ellipse and related results

There are 2 questions(part of same question but I divided it into two): Q1. Prove that the length of the focal chord of the ellipse $\frac {x^2}{a^2}+\frac {y^2}{b^2}=1$ which is inclined to the ...
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1answer
26 views

Plotting 3 equidistant points on a sphere

.Hello! I'm trying to figure out how to plot with x,y,z, three points that are equidistant along the surface of a sphere from each other that are all on a horizontal axis (so y = 0) with a radius of ...
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1answer
14 views

Hyperbola / Rotated Hyperbola Intersection

I am trying to find the point where two hyperbolas intersect, that is, to find a vertex that is common to both hyperbolas. Also, note that I am only testing for a region of both hyperbolas -- only a ...
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3answers
41 views

The locus of points $z$ which satisfy $|z - k^2c| = k|z - c|$, for $k \neq 1$, is a circle

Use algebra to prove that the locus of points z which satisfy $|z - k^2c| = k|z - c|$, for $k \neq 1$ and $c = a + bi$ any fixed complex number, is a circle centre $O$. Give the radius of the circle ...
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1answer
46 views

Motivation for constructing $F$ s.t. $\ker(\text{curl}) \subset \text{Im}(\text{grad})$, $\ker(\text{div}) \subset \text{Im}(\text{curl})$

In 'from calculus to cohomology', we consider the space $V$ of smooth functions $U \to R^3$, with $U \subset R^3$ star-shaped (i.e. convex), and for cohomology reasons (showing $H^1(U)=H^2(U)=0$) we ...
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1answer
28 views

Area of surface parametrized in spherical coordinates

Suppose we have a smooth, bounded, closed surface in $\mathbb{R}^3$ which can be parametrized by giving the distance from the origin as a function $r(\varphi,\theta)$ of spherical angles ...
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3answers
98 views

3D coordinates of circle center given three point on the circle.

Given the three coordinates $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$, $(x_3, y_3, z_3)$ defining a circle in 3D space, how to find the coordinates of the center of the circle $(x_0, y_0, z_0)$?
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2answers
26 views

Solving for unknowns in parametric equation

I have the parametric equation of a circle: $$f(u) = \langle a \cos(u) + b, a \sin(u) + c\rangle,$$ and because the equation has $3$ unknowns $a,b$ and $c$, I have been given $3$ points $p_0, p_1$, ...
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1answer
103 views

Finding the points of intersection of a circle and a line

In a test (of math in arabic language) we we're asked to find the points of intersection of a circle and a line. Their equation is given. In the test I solved system of equations made of their ...
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1answer
94 views

Compute the area of a parallelogram defined by a particular construction

I got stuck with this mathematical task. Can someone help me how to solve this problem? I need to find the F(area) value. It is kind of a thinking task Context The problem is extracted from a ...
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1answer
51 views

Standing at the center of a cube and walking halfway to a wall - field of vision

In my python programming class one of the bonus problems is this: Suppose you are located at the exact center of a cube. If you could look all around you in every direction, each wall of the cube ...
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1answer
33 views

Find the equation of parabola tangent to a line

I know how to find the equation of the line tangent to a parabola through a certain point. But how do I find the equation of the parabola from the point and the tangent line? For example, how do I ...
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1answer
23 views

Recurrence relation of distances between $n$-dimensional curves

I have a problem involving recurrence and euclidean distances in $n$-dimensional curves. Given the sequence of curves in $\mathbb{R}^n:$ $\{ x_{1}^2+x_{2}^2+\cdots + x_{n}^2 = 1, ...
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1answer
36 views

Calculating XY coordinates on line

I have been working on this problem for a while now and can’t figure out the solution. Hence my post on this forum. I’m trying to figure out the position of a symbol on a line. These lines are located ...
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0answers
37 views

intersecting point of two lines

The circle has R radius and and ellipse is intersecting the circle. I need to findout $x_c$ and $y_c$, which is the midpoint of the 2 intersected point of ellipse.Line 3 is the tangent of the ...
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1answer
27 views

How do I approach this geometrical problem?

For a point $P=(x,y)$ write $f(P)=ax+by$. Let $f(A)=f(B)=10$. $C$ be a point not lying on the line joining $A$ and $B$. $C^{'}$ be the reflection of $C$ w.r.t. this line. If $f(C)=15$, find ...
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1answer
19 views

find x in coordinates given the angle

This is the problem: if the angle from the line through $(-4,2)$and $(3,-4)$ to the line through $(-4,2) (x,3)$ is arctan 37/29 find the value of $x$? Should i use this formula: $$\tan \theta= ...
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1answer
29 views

defining a closed curve in cartesian coordinates

I am trying to implement a track in cartesian coordinates, such that X and Y coordinates are accepted and those are linearly interpolated. The problem is, I want to include circular shapes on ...
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1answer
29 views

Coordinates of a vertex of a triangle?

Here is the problem: There is a triangle with vertices $A,B,C$ in a cartesian coordinate system, where coordinates of points $A$ and $B$ and the angle $\alpha=\measuredangle ABC$ are given. The ratio ...
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0answers
20 views

Simplest way to calculate the width of a segment of a convex shape

A convex shape $C$ is cut using a a chord. What is the width of the resulting segment? This is the length of the green thick short line in the figure below: Here is my current solution: Mark the ...
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0answers
34 views

Finding the point on an ellipse most distant from a given line

$\mathrm C$onsider an ellipse with the origin as its centre, i.e., of the type $$\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1$$ and a line joining two points on the ellipse. $\mathrm T$he problem is to ...
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2answers
62 views

Intersection of a cone and a plane.

I need a proof that the intersection of a cone with a plane parallel to the cone's axis is a hyperbola.
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1answer
53 views

How do you 'rotate' a polynomial?

I have a polynomial equation: $$y=(-5 \times 10^{-6} \times x^3)+(0.0004 \times x^2)+(0.0582 \times x)-0.4397$$ Is it possible to "rotate" this polynomial curve (maintaining the shape) around the ...
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1answer
22 views

Find the images of (1,0) under reflection in L?

Consider the line $$L = \{(x,y): x - 2y = 2\}$$ Find the images of $(1,0)$ under reflection in $L$? Thanks in advance.
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2answers
60 views

Indefinite integral with sector of ellipse

An ellipse is given by the following equation: $$ 152 x^2 - 300 x y + 150 y^2 - 42 x + 40 y + 3 = 0 $$ After solving for the midpoint we have: $$ 152 (x-1/2)^2 - 300 (x-1/2) (y-11/30) + 150 ...