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Questions tagged [analytic-geometry]

Questions on the use of algebraic techniques for proving geometric facts.

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How do you find a locus of the y coordinate of a coordinate pair in an isosceles triangle?

I need to find the possible values for a y coordinate of a point in a isosceles triangle. How do I figure out the values of y which will also satisfy the rules of an isosceles triangle?
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1answer
34 views

How can line segments that don't meet be called “perpendicular”?

Suppose there is a line segment from $(4,0)$ to $(6,0)$, and another line segment $(0,1)$ to $(0,2)$. They don't form an angle, so how are they "perpendicular"? What actually is meaning of it? Like ...
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Polar equations of conics versus Logistic growth

I noticed that two equations have very similar forms. The polar equation of a conic section with focus at the pole, eccentricity$=e$ and with directrix given by the line $x=d$ is: $$r=\frac{ed}{1+e\...
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Flexible grid update algorithm

I have a 2D square grid where the edges are line segments and the vertices have moved from their initial positions. Based on external constraints, the vertices are submitted to further motion, under ...
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2answers
21 views

Volume of the pyramid - how to find the coordinates of 4 vertices?

The pyramid is bounded by planes: $x=0$, $y=0$, $z=0$ and $9x-y-3z=54$. IT is needed to calculate the volume $V=\frac {1} {3} bh$, where where $b$ is the area of the base and $h$ the height from the ...
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1answer
43 views
+50

Three statements regarding a normal and a tangential plane

I am new to calculus, and was given the following question to answer. I have worked out an answer, but am not 100% sure about the details. Any feedback would be great! Many thanks in advance. Given ...
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1answer
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Finding the perpendicular line of a given line.

How can I find the line that contains the point $\ (0,2,1) $ and intersect the line $\ (2t,1-t,2+t) $ in $\ 90 $ degrees Maybe since the direction vector of the given line is $\ (2,-1,1) $ and the ...
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1answer
55 views

Circles, coordinate geometry

If P and Q are the points of intersection of the circles$$ x^2 + y^2 + 3x + 7y +2p – 5= 0$$ and $$x^2 + y^2 +2x + 2y – p^2 = 0$$ then there is a circle passing through P, Q, and (1, 1) for ... a)all ...
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3answers
67 views

What is the value of $|\alpha|$?

Let the complex numbers $\alpha$ and $\frac{1}{\bar{\alpha}}$ (notice the bar above $\alpha$) lie on the circles $(x-x_0)^2+(y-y_0)^2=r^2$ and $(x-x_0)^2+(y-y_0)^2=4r^2$ respectively. If $z_0=x_0+iy_0$...
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Calculate ellipse eccentricity [on hold]

There are any equation for calculate ellipse eccentricity only with major and minor axis parameters? Thanks in advance!
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Circumference touching a sine wave

I'm trying to get the intersection points between a sine function and a circumference. So, i have this equations: $y = a\sin(bx + c) + d$ $(x-h)^2 + (y-k)^2 = r^2$ If i substitute the sine in the ...
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1answer
35 views

Distance of the point $(a,b,c)$ to the plane $z=0$

I'm trying to solve a calculus problem, I need to find the mass of a cylinder, I'm close to the answer, I got $8\pi$ but it should be $16\pi$. I think my mistake lies in the density function since it ...
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1answer
31 views

Finding and minimizing the length of a string wrapped around a cylinder.

A string of length $l$ is wrapped around a cylinder of diameter $d$ and height $h$. The string does $n$ turns and starts at one end of the cylinder, ending at the top. The pitch of the resulting ...
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How to find the value of the projection of an angle onto a plane?

Consider a triangle (T) in 3D space of given vertices A, B, and C. A given ray (R) (assumed to be in the direction of the x-axis) hits the triangle at one of its vertices - say A. Let $\theta$ be ...
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what does it mean that plane XY divides the line joining (2,4,5) & (3,5,-4) with a negative ratio

What does it mean that plane $XY$ divides the line joining $(2,4,5) \text{ and } (3,5,-4)$ with a negative ratio? It's an old question I saw when I was in high school. Search results didn't help. I ...
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3answers
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5 numbers are enough to give a line

My question is very elementary; I just want to ask if it is widely known (probably yes) and whether this is written in textbooks (where). A line in the 3-dimensional space is usually given either by ...
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3answers
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What's the condition for a plane and a line to be coplanar in 3D?

Please correct me if I'm wrong, but given a plane expressed in point-normal form, and a line expressed in parametric form, an easy way of finding the intersection (or lack of it) is substituting the $(...
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2answers
38 views

Complex coordinates of the vertices of a polygon

If $z_0$ be the centre of a regular polygon of $n$ sides and $z$ be its one vertex $A_1$, then the vertices $A_2, A_3,\dots, A_n$ (proceeding in anticlockwise direction, taking actual position of ...
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4answers
165 views

Distance Formula Problem

If two vertices of an equilateral triangle are $(1, -1)$ and $(-\sqrt{3}, - \sqrt{3})$, find the coordinates of the third vertex. Step by Step procedure to get the answer. Take $A=(1, -1)$, $B=(-\...
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1answer
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Is there a formalization of the link between geometry and analytical geometry?

Geometry and algebra/calculus can be formalized by axioms. Is there a global theory that combines both and establishes correspondences such as the equation of a straight line is $ax+by+c=0$, the ...
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Analogous of Power of Point in Euclidean Geometry in high dimension

While playing around with dot product in 2D, I realized it's scalar projection behavior is directly related to power of point in Euclidean Geometry. I am wondering if there is any notion similar to ...
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How to fill a rectangle with smaller specific rectangles that have cardinal information about their adjacent neighbours

Lets say its 6x6 grid that is represented by top left(0,0) and bottom right(1,1) in coordinate system. Next, I have set of objects with their cardinal directional information about each of their ...
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78 views

Planar sines are special cases of spherical sines?

In the figure $BC$, $AC$ and $BA$ are all large arcs, the spherical angle $∠ACB$ is right angle, and $\Delta ABC$ is spherical right triangle. $O$ is spherical center. Let the radius of the sphere $...
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1answer
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Analytic geometry reference request?

Do you know any rigorous introduction to analytic geometry? I'm looking for a reference which defines vectors as equivalence classes of oriented segments. Thanks.
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Check if from given $N$ vertical line segments and adding some horizontal line segments closed shape can be formed

We have given $N$ vertical line segments each of the form $X_i, Y_{1i}, Y_{2i}$, it is guaranteed that the set is valid, i.e. no two line segments intersect. We need to check if we can add some more ...
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1answer
24 views

Finding a line with the following criteria

Find the line that goes through $\ (0,1,2) $, is parallel to the plane $\ x+y+z-2=0 $, and is perpendicular to $\ r(t) = (1+t,1-t,2t) $. I understand that the line is perpendicular to the vectors $\ (...
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1answer
43 views

Determine if a point lies in a quadrangle [duplicate]

I have a quadrangle which sides consist of parts of rays, and I only know the coordinates of two points on each ray. I need to determine if a point $(x,y)$ lies in such quadrangle. In this picture, ...
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1answer
45 views

How to find the equation of a hyperbola given the asymptote, equation of axis and a point

Given that a hyperbola has asymptote $y=0$, passes through the point $(1,1)$ and has axis $y=2x+2$, determine its equation. The answer arrived at is $\displaystyle{4xy+3y^2+4y-11=0}$. However, I have ...
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1answer
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How to find slope of angle bisector of two lines with slope M1 and M2? [closed]

Given two slopes(M1 and M2) of two distinct lines, is there any way to find the slope(M3) of angle bisectors of those two lines?
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2answers
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What is the probability that a randomly chosen point lies inside a given parabola?

A parabola (say $y^2=4ax$) divides the coordinate plane into two regions: one considered to be "inside" the parabola and the other "outside". However, both these regions are infinitely large. What is ...
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$P$ is at constant distance $2$ from point $(3,5)$. Find the equation of the locus of $P$.

The question states: P is at a constant distance of two units from the point (3,5). Find the cartesian equation of the locus of the set of points P in each case. To solve this I drew it out, but ...
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2answers
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minimum of a $MA^2+MB^2$

Having the point $A(8,0),B(0,6)$ and $M(x,y)$ and arbitrary point in plane.I have to find the minimum of $MA^2+MB^2$? Can somebody give me some tips, please?
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2answers
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The intersection points of an ellipse with a line parallel to the line which contains the two foci of the ellipse, and the foci are concyclic.

I was playing around with ellipses in a graphing app and I found this property which seems to be true, but I can't find a way to prove it. Let A and B be the foci of the ellipse, draw a line parallel ...
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1answer
45 views

Finding the singular locus of the given complex space

This problem is from Greuel et al., Introduction to Singularities and Deformations. Determine the singular locus of the complex spaces defined by the following $\mathcal{O}_{\mathbb{C}^n}$-ideals: (...
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how can I show that the condition that the curve $u(x,y,z) = 0$, $v(x,y,z) = 0$ should touch the surface $w(x,y,z) = 0$

how can I show that the condition that the curve $u(x,y,z) = 0$, $v(x,y,z) = 0$ should touch the surface $w(x,y,z) = 0$ is that the eliminant of $x$, $y$, and $z$ from these equations and the further ...
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1answer
38 views

Finding a point on a circle in 3D space which is a given distance away from another point

I am trying to find a point, lets call it $X = (x_1, x_2, x_3)$, on a circle in 3D space (with a center $C_1$, radius $r$ and unit vector $\vec{AX}$ perpendicular to the plane of that circle ) which ...
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Mapping between planes with a polynomial function

While mapping between two planes, say the mapping function is a polynomial. The planes are BOTH rectangles! The first plane is mapped onto the second plane in such a manner: X=Function1[x,y,z]= ...
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2answers
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angles in complex plane

There are two points $z_1$ and $z_2$ in the complex plane. What is the angle that the line segment $z_1 z_2$ subtends at the origin? I want to find this angle in terms of $z_1,z_2$ and $|z_1−z_2|$ ...
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Finding quaternion, representing transformation from one vector to another [closed]

Intro. Previously, I've asked a question on converting rgb triple to quaternion. After that question I've managed to get unit quaternions. Since they were unit ...
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3answers
32 views

Find equation of a circle through point that touches two given lines

We have given point of a circle $A(0, 0)$ and two lines: $x+y+2=0 \text { and } x-y+4=0$. We should find equation of a circle that passes through the point $A$ and touches those two lines. I started ...
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Question on finding cartesian equation of a plane(general equations of a plane)

Given the equation of a plane $5x+2y-z+22=0$ ,Find the vector $\vec V$ perpendicular to the plane and point $P_o(x_o,y_o,z_o)$ on the plane ? Trying Solutions: Since the general equation is $Ax+By+...
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1answer
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Deducing the locus of a point of intersection of two lines.

A straight line $L$ through origin meets $x+y=1$...$(1)$ at $P$ and $x+y=3$...$(2)$ at $Q$. Through $P$ and $Q$ two lines $L_1$ and $L_2$ are drawn which are parallel to $2x-y=5$...$(3)$ and $3x-y=5$.....
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1answer
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the triangle cosine relation in a complex plane

I have a triangle $ABC$ in a complex plane. The arrangement of vertices is in a counterclockwise direction. The coordinates of $A$,$B$,$C$ are $z_A$,$z_B$,$z_C$ respectively. It is given that length ...
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2answers
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What's the geometric interpretation of this “vector cross product”?

This answer on StackOverflow answers a question about intersection of two segments. Right at the beginning, it introduces a “vector cross product”. Define the 2-dimensional vector cross product $\...
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Question about Helmholtz's paper “ON THE CONSERVATION OF FORCE”

Below follows the exact extract from Helmholtz's paper "On the conservation of force". Let us now imagine, instead of the system $A$, a single material point $a$, it follows from what has been just ...
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1answer
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The concurrency of the heights of a tetrahedron [duplicate]

The sentence says that the opposite edges of the tetrahedron ABCD are perpendicular and the fact that $AB^2+CD^2=AC^2+BD^2=BC^2+AD^2$ and I need to prove that the heights of the tetrahedron are ...
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Communication on the boundary of a $C^1$ domain

Assume $\Omega$ is a $C^1$ bounded domain of $\mathbb{R}^d$, $d \geq 2$. For $(x,y) \in (\partial \Omega)^2$, we say $x \sim y$ if $n_x \cdot (y-x) > 0$, $n_y \cdot (x-y) > 0$, and $(tx+(1-t)y) \...
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2answers
67 views

Equation of plane containing a point and a line

Find the equation of the plane containing the point $A(0,1,-1)$ and the line $(d) : \begin{cases} 2x - y + z + 1 = 0 \\ x + y + 1 = 0 \end{cases}$ Where should I start? I was thinking about writing ...
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3answers
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Three normal to the parabola $y^2=x$ are drawn through the point .

Three normal to the parabola $y^2=x$ are drawn through the point $(c,0)$ then $$\textrm {a}. c=\dfrac {1}{4}$$ $$\textrm {b}. c=1$$ $$\textrm {c}. c>\dfrac {1}{2}$$ $$\textrm {d}. c=\dfrac {1}{2}$$...
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Barycentric and projective coordinates

I have a question: what is the relation between the barycentric and the projective coordinates? Are the first one a particular case of the second? Thank! Edit: the setting is the plane, and in ...