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

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2
votes
2answers
156 views

Area of ellipse not in xy-plane

I've got a problem in which I'm trying to find the area of an ellipse which is given by the intersection of an elliptic cylinder with a plane. Nothing here is parallel to the coordinate axes, which is ...
0
votes
1answer
151 views

Finding equation of ellipse with given point and distance between directrices

I need to find the equation of an ellipse. The given were just a point where it passes, and distance between directrices. I know that the distance between directrices is given by $2a/e$. I don't ...
0
votes
1answer
44 views

Find the equation of a hyberboloid with given base, narrowest section, and the distance between them

I have one question left in an assignment and I havn't been able to solve it. I know the equaton for a hyperboloid and I know that $a$ and $b$ will be equal to each other. I don't know how to solve ...
1
vote
3answers
71 views

Finding circle of a sphere through two points

We have two points $P_1, P_2$ on a sphere $S$ of radius $R$. Suppose for $r \ll R$, the distance between $P_1$ and $P_2$ is less than $2r$. Then, $P_1$ and $P_2$ both lie on exactly two radius-$r$ ...
1
vote
1answer
354 views

Intersecting two parabolas and computing the angle between the tangents in a point of intersection

I was solving some problems on parabola. I saw a question and solved it, but my solution was way too big. The question was: If $$\left(\frac{a}{b}\right)^{1/3}+\left(\frac{b}{a}\right)^{1/3} = ...
1
vote
0answers
189 views

Proof 5 points determine a conic without projective geometry

So I'm trying to prove that any five points, of which no 3 are colinear, there is a single conic that passes through al of them. I don't want to use projective geometry but rather, only analytic ...
1
vote
4answers
1k views

How to tell whether a point is to the right or left side of a line

I have a line equation in the form ax+by+c=0 and a point p(x,y).How can I determine on which side of the line the point is located?
0
votes
1answer
24 views

Graphing and finding sine wave info with $\sin(x/4)$

I'd doing a chapter on graphing sine waves, and finding the amplitude, period, and so on. I know something like $y = 2 \sin(3x+ \pi) + 1$ can be turned into $y = 2 \sin[3(x + \pi/3)] + 1$ following ...
0
votes
2answers
55 views

How to connect a line between 4 randomly placed points on a plane such that the line does not cross itself

You get 4 coordinates of points on a plain. You need to connect them all with a line. The line must not cross itself. What's your strategy?
0
votes
1answer
83 views

Equation of a parabola: Translations and rotation

I've tried to solve this problem: Find an equation of the parabola with vertex at point $(1,1)$ whose directrix is the line $x-2y=6$. It has to be solved using translation and rotation (coordinate ...
2
votes
1answer
85 views

Find the distance of a point from a plane generated by two given vectors

I need to calculate the distance of the point $P = (0, 5, -4)$ from the plane which pass from the point $P1=0, 1, -2)$ and generated by the two vectors: $$ v1 = (1, 2, 3), v2 = (-1, \sqrt{2}, 1) $$ ...
0
votes
2answers
130 views

What is the minimum information required to define an equation for ellipse?

What is the minimum information ie. amount of points in 2-dimensional plane in order to define the equation for an ellipse? I know that unique ellipse cannot be defined when only one of the foci is ...
0
votes
2answers
94 views

What is the best book for coordinate geometry?

Requirements: A) not too thick , as I am reading this only to solve calculus problem . B) free on web is the best (optional) C) I don't mind if the book involves both coordinate geometry and ...
1
vote
2answers
62 views

Can ellipse equation be transformed through one of its foci?

Can we transform ellipse equation to represent an ellipse transformed by tilting it through its focus such that its center point moves in circular manner and one of its focus stays at constant ...
1
vote
2answers
179 views

Is it possible to find equation for ellipse when focus, eccentricity 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 ...
0
votes
1answer
84 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 ...
2
votes
0answers
18 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 ...
0
votes
2answers
22 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
2answers
66 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 ...
1
vote
2answers
62 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
19 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
57 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 ...
2
votes
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 ...
0
votes
2answers
50 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 $\sin(\alpha)+\sin(\beta)\ge 1$?
4
votes
1answer
85 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
62 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
34 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 ...
0
votes
1answer
126 views

How to find the length of the focal chord making angle $\theta$ with the axis of parabola?

A focal chord of $Y^2 = 4aX$ makes angle $\theta$ with the axis of the parabola. How can I find the length of the chord? I have used the parametric equation but couldn't go further.
8
votes
4answers
321 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
83 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
41 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
988 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 ...
0
votes
1answer
56 views

How do I place two points on two axis-aligned line segments such that they have given Euclidean distance l?

This is my problem: I am given two axis aligned line segments $l_1$ and $l_2$ of finite length and a distance $l$. How do I find points $p_1 \in l_1$ and $p_2 \in l_2$, such that $||p_1-p_2||_2 = l$? ...
2
votes
6answers
149 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
933 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
278 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
38 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 ...
4
votes
2answers
461 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
80 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
152 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
48 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
68 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
50 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 ...
4
votes
5answers
1k 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 $(x_0, y_0, z_0)$?
1
vote
2answers
48 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
182 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
166 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
276 views

Calculate the distance between intersection points of tangents to a parabola

Question Tangent lines $T_1$ and $T_2$ are drawn at two points $P_1$ and $P_2$ on the parabola $y=x^2$ and they intersect at a point $P$. Another tangent line $T$ is drawn at a point between $P_1$ ...
4
votes
1answer
111 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
106 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 ...