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

learn more… | top users | synonyms (1)

0
votes
2answers
14 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 ...
0
votes
0answers
19 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 ...
0
votes
2answers
22 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 ...
0
votes
1answer
22 views

Simple Analytic Geometry Problem [on hold]

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
0
votes
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 ...
2
votes
0answers
13 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 ...
0
votes
0answers
25 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 ...
-1
votes
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 ...
1
vote
2answers
20 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 $x=1$.
1
vote
0answers
43 views

finding the value of $a$ [closed]

Given the function $y=\frac{ax+5}{x^2-ax-6}$ with extremum at $-5<a<0,0<a<5$. Through the extreme points we build lines that are Parallel to the axes, so that The result is a rectangle ...
2
votes
1answer
17 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 ...
0
votes
2answers
46 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 $\sin(\alpha)+\sin(\beta)\ge 1$?
1
vote
1answer
41 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 ...
1
vote
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 ...
2
votes
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 ...
6
votes
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 ...
1
vote
1answer
19 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 ...
1
vote
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 ...
0
votes
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 ...
2
votes
6answers
83 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 ...
2
votes
3answers
75 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.
0
votes
2answers
40 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?
0
votes
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 ...
3
votes
1answer
24 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 ...
0
votes
1answer
25 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 ...
0
votes
1answer
12 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 ...
3
votes
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 ...
3
votes
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 ...
1
vote
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 ...
3
votes
3answers
94 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 $(x_0, y_0, z_0)$?
0
votes
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 $p_0, p_1$, ...
1
vote
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 ...
2
votes
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 ...
4
votes
1answer
50 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 ...
0
votes
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 ...
0
votes
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, ...
1
vote
1answer
35 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 ...
0
votes
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 ...
-1
votes
1answer
63 views

What is the value of $k$? [closed]

A line has an equation of $x+5y-4=0$. If the line $x-ky-11=0$ makes an angle of $45^\circ$ counterclockwise from $x+5y-4=0$, find $k$. (If someone knows the answer, please tell me what it is and how ...
0
votes
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 ...
0
votes
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= ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
2answers
61 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.
0
votes
1answer
51 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 ...
0
votes
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.
1
vote
2answers
58 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 ...
0
votes
1answer
30 views

doubt with direction angles

Is it possible for a 3D vector to be drawn with the direction angles of $\alpha=45^\circ$ and $\beta=45^\circ$ ? if yes what is measure of $\gamma^\circ$? I calculated $\cos^2(45^\circ ...