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

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2
votes
1answer
70 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
44 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
23 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
22 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
34 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
29 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 ...
0
votes
2answers
32 views

Line passing through given points [closed]

Calculate the equation of the line through the points $(-1, 1)$, $(4, 0)$, and $(24, -4)$. the choices of answers are $y=-\dfrac15x+4/5$ $y=\dfrac15x+4$ $y=-5x+4$ $y=5x+6$
-2
votes
1answer
49 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
25 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
16 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
0answers
42 views

Find the slope given the angle [closed]

This is the problem: find the slope of the Line 2 such that the tangent of the angle from line 1 to line 2 is -1/2? I already use the formula of $$\tan= ...
0
votes
1answer
27 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
26 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
18 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
29 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
53 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
46 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
21 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.
0
votes
2answers
53 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
25 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 ...
0
votes
0answers
18 views

Proof in analytic geometry using vector multiplication

Let us have a triangle $\Delta ABC$. $H$ is the intersection of heights if and only if $$ \overrightarrow{HA}\cdot \overrightarrow{HB} = \overrightarrow{HB}\cdot \overrightarrow{HC} = ...
0
votes
0answers
32 views

Proof in analytic geometry

Let us have a triangle $\overset{\Delta}{ABC}$. $H$ is the intersection of heights if and only if $\vec{HA}.\vec{HB}=\vec{HB}.\vec{HC}=\vec{HC}.\vec{HA}$ I think that it has to deal with vector ...
1
vote
1answer
26 views

Algebraic step on a trig expressiom in linear algebra

$$W = ||V||(\cos(\varphi)\cdot \cos(\theta) - \sin(\varphi)\cdot\sin(\theta), \cos(\varphi)\cdot\sin(\theta) + \sin(\varphi)\cdot\cos(\theta))$$ $$= (v_1 \cos(\theta) - v_2 \sin(\theta), v_1 ...
1
vote
0answers
58 views

Find max distance from $(0,0)$ to line defined on ellipse.

I have got a following problem : $E = \{ \frac{x^2}{a^2} + \frac{y^2}{b^2} =1 \}$ $N$ - line (normal) perpendicular to E at $(x_0,y_0)$ Find max $dist(N,(0,0))$ So I am starting with attempt to ...
0
votes
2answers
37 views

Questions about elipse

Given the center of an elipse and three of its points, is this elipse completely determined? What is the easiest way to show that five points of an elipse are enough to determine the elipse?
2
votes
2answers
27 views

Find intersection points of a line with a circle, and the equation of another circle passing through those points [closed]

If the line $x=2y$ meets the circle $x^2+y^2-8x+6y-15=0$ at points $P,Q$, find the co-ordinates of $P$ and $Q$ and the equation of the circle passing through $P,Q$ and at the point $(1,1)$. Could I ...
1
vote
1answer
21 views

How to find a point at a certain distance to other points on the same line

Assuming the points A(x1,y1) and B(x2,y2) and distances between AB (d1) and AC (d2) are known. How can I find the point C(xp,yp)? Actually it has a trivial solution, writing the distance equation 2 ...
0
votes
2answers
21 views

Find a Cartesian Equation for the Plane Satisfying Those Properties

Find the Cartesian equation of the plan parallel to j and passes through the intersection of the planes described by the equations x + 2y + 3z = 4, and 2x + y + z = 2. I was able to get the ...
0
votes
0answers
16 views

What is the analog of the scalar triple product in two dimensions?

Is there a standard name and/or a notation for the analog of the scalar triple product in two dimensions? Namely, i am interested in the following operation: given two elements $\vec u$ and $\vec v$ ...
3
votes
1answer
67 views

Triangle, Circle Problem

What is the area $\triangle DEF$ ? I solved it using analityc geometry. I want to see if there is way to solve it using plane geometry. I did it: $x^2+y^2=400$ $(x+10)^2+y^2=100$ I found the ...
0
votes
0answers
15 views

Plane and symmetrical lines

I need to solve this problem (sorry for bad english) I have plane $\pi$ and line $p_1$ intersecting $\pi$ in point $P$. Then I find line $p_2$ symmetrical to line $p_1$ where $\pi$ is plane of ...
0
votes
1answer
21 views

Locus of intersection between $y= 8\lambda/(\lambda ^2 + 4)$ and $y =2 \lambda x/(4-\lambda^2)$

I have the equations $$y=\frac{4\lambda}{\frac{1}{2}\lambda^2+2}\quad \text{and}\quad y=\frac{\lambda x}{-\frac{1}{2}\lambda ^2 + 2}$$ each representing a line. I'm asked to find the locus of the ...
1
vote
2answers
27 views

How to interpret the equation of a line in 3D through two points, when there are $0$s in the denominator? [closed]

If $A=(0,0,0)$ and $B=(1,0,0)$ are two points of a line in three dimensions, I think its equation should be $$\frac{x-0}{1}=\frac{y-0}{0}=\frac{z-0}{0}\tag1$$ according to the formula ...
0
votes
1answer
21 views

How to show existence of an orthogonal map?

I want to show that the following holds: Let $x,y\in \mathbb{R}^n\setminus\{0\}$ be given and such that $\|x\|=\|y\|$. There is an orthogonal map $T$ such that $Ty=x$ (a rotation). How could one ...
9
votes
5answers
355 views

Circle radius as variable

I am confused. How is $y^2 + x^2 =3x$ a circle? Can someone please help me try to understand why the above a circle, or is it just a typo?
0
votes
2answers
23 views

Find the equation of the plane that contains:

Find an equation for the plane containing the lines $$x = 5y = \frac{z + 1}{4}$$ and $$\begin{cases} x = t \\ y = 2t\\ z = 6t − 1 \end{cases}.$$ I know that finding two points will allow me to find ...
2
votes
1answer
44 views

Finding a line through 4 other lines!

This one's probably easy, but I'm dreadfully stuck and can't seem to figure out a decent method. I have the following lines: $$a: \vec{x}(\lambda)= \left( \begin{array}{ccc} 4 \\ -2 \\ -2 ...
2
votes
2answers
23 views

Showing that normal line passes through a point.

I need to show that a line passes through a point. How should I go about doing this? The question is: Let $L$ be the normal line at $(1,1,1)$ to the level surface of $f(x,y,z) = x^2 - z$ that ...
0
votes
0answers
11 views

Equivalence of definitions for a conic

I have to prove that these two definitions for the eccentricity of a conic $C$ are equivalent: Ratio between the distance of the points $x$ in $C$ to $f$ its foci and $l$ its directrix. Ratio ...
2
votes
2answers
43 views

What is the Equation for a straight line in a 3D space? And how to find other parelell lines to it?

We know $y=mx+c$ is the equation for a straight line in a 2D graph. And the parallel line that goes through $(x_1,y_1)$ is $y=mx+(-mx_1+y_1)$. But how do we display the straight line in a 3D graph ...
0
votes
4answers
54 views

Show that if an ellipse and a hyperbola have the same foci, then at each point of intersection their tangent lines are perpendicular.

I have to show that: If an ellipse and a hyperbola have the same foci, then at each point of intersection, their tangent lines are perpendicular. So I know that if I prove it for one of the ...
3
votes
1answer
62 views

Is every smooth $\mathbb{R}$-variety isomorphic to an affine variety?

I sadly don't know anything about formal GAGA yet, but I am at least trying to follow my intuition as often as possible. In differential geometry we know that we can embedd every smooth ...
0
votes
2answers
84 views

Smallest circle enclosing three disjoint circles

Consider three disjoint circles not necessarily of same radii. How do you draw the smallest circle enclosing all these three circles? Where is its centre, and what is its radius?
1
vote
0answers
11 views

Finding coordinates of ground-zero with seismic sensors

At the unknown t0 time an explosion occurred at an unknown point X,Y on the 2D plane. We ...
0
votes
1answer
43 views

Equidistant points on a circle

I would like to obtain/generate points on a circle in Cartesian coordinates such that the distance between two consecutive points will be always equal. For example, plotting a circle with radius 100 ...
2
votes
0answers
52 views

how to find angle between two added up vectors in cartesian space

I would like to find the angle between two vectors (theta) -> v1 From i to i+1 v1=(xi1-xi , yi1-y1) and v2 from i+1 to i+2 v2=(xi2-xi1, yi2-yi1), which are shown as in the figure (but v1 and v2 can be ...
2
votes
4answers
95 views

How to find coordinates of reflected point?

How can I find the coordinates of a point reflected over a line that may not necessarily be any of the axis? Example Question: If P is a reflection (image) of point (3, -3) in the line $2y = ...
0
votes
1answer
19 views

help needed in understanding general conics proof

The origin is a centre of a general conic of second degree iff the coefficients of linear terms vanish. $ (\Rightarrow)$ part: Let $$ Q(x,y)\equiv ax^{2}+2h xy+ by^{2} + 2gx+2fy+c=0$$ books ...
4
votes
2answers
63 views

Coordinates of the intersection of two tangents to a circle

Let $A = (x_A, y_A)$ and $B = (x_B, y_B)$. Let $\gamma$ be a circumference of radius $r$, centered in $(0, 0)$; $A$ and $B$ lie outside of $\gamma$, and on the same side of some line $L$ through the ...
3
votes
1answer
37 views

What is the number of intersections of diagonals in a convex equilateral polygon?

Question: [See here for definitions]. Consider an arbitrary convex equilateral polygon with $n$-vertexes ($n\geq 4$) and the $n$-sequence $\langle \alpha_i~|~i<n\rangle$ of its angles which ...