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Questions tagged [analytic-geometry]

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

3
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1answer
55 views

rational points on the quadrifolium $(x^2 + y^2)^3 = (x^2 - y^2)^2$

I have been reading the Wikipedia page on the Quadrifolium there are two of them: \begin{eqnarray*} r &=& \sin 2\theta \\ (x^2 + y^2)^3 &=& 4 x^2 y^2 \end{eqnarray*} and it's $45^\...
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0answers
8 views

Is it possible to estimate 3D coordinates from 2D coordinates from a single image? [on hold]

I want to estimate 3D coordinates to solve occlusion problem from 2D coordinates using a single RGB image. Is it possible or not?
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0answers
27 views

How do I find the vertices, foci, eccentricity, and the lengths of the minor and major axes of the following ellipse?

How do I find the vertices, foci, and eccentricity of the following ellipse? $$\frac{x^2}{49} + \frac{y^2}{25}=1$$ I put the ellipse in the standard equation so far, but am not sure what to do in ...
3
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3answers
43 views

Show that two cardioids $r=a(1+\cos\theta)$ and $r=a(1-\cos\theta)$ are at right angles.

Show that two cardioids $r=a(1+\cos\theta)$ and $r=a(1-\cos\theta)$ are at right angles. $\frac{dr}{d\theta}=-a\sin\theta$ for the first curve and $\frac{dr}{d\theta}=a\sin\theta$ for the second ...
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0answers
5 views

Prove that a harmonic homology preserves the conic

I came across a question in the book by Judith N. Cederberg and I’m learning about projective geometry. One of the question was “Show that a harmonic homology whose centre and axis are pole and ...
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0answers
24 views

How to determine what kind of conic section in the affine plane?

So, I've been struggling a bit with understanding this problem. Let $P^2$ be the real projective plane with homogenous coordinates $(x_0:x_1:x_2)$ Let $\cal{C}$ be the line given by $$x_0^2 + 2x_0x_1+...
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1answer
33 views

Determine the polarities of a self polar triangle

Consider a triangle $PQR$, $P(0,2,1), Q(1,0,2), R(0,4,9)$. Determined the polarities if triangle $PQR$ is self polar. By definition of self polar triangle, point $P$ gets mapped to line $QR$, point $...
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1answer
21 views

How to find the domain of a function using spherical coordinates?

Hey guys so I was wondering how to find the domain of a function using spherical coordinates.. For example, I take two functions: F(x,y,x)=√x+√y+√z+ln(4-x²-y²-z²) whose domain is D=((x,y,z):x≥0,y≥...
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2answers
25 views

Find tangets to a ellipse not centered at(0,0) that pass through point P

I must find the two tangents that pass through the point $(2,7)$ for the ellipse $2x^2+y^2+2x-3y-2=0$ all I was able to was getting $\frac{dy}{dx}=\frac{(-4x-2)}{(2y-3)}$ and therefore equalizing ...
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1answer
28 views

Find the value of k, so that the following lines intersect at the same point [on hold]

Find the value of k, so that the following lines intersect at the same point: $$3x + y - 2 = 0$$ $$kx + 2y - 3 = 0$$ $$2x - y + 3 = 0$$ How can I resolve this? thanks I was able to find that $(-\...
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0answers
8 views

Two rotations of different axes

If we know that any rotation of $SO(3)$ can be disturbed as a product of two specular symmetries and that one of them can be arbitrarily chosen between the planes that contain the axis of the rotation....
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0answers
83 views

Finding the maximum of sum

Given a right hexagonal pyramid $SA_1A_2…A_6$. $A_1 = (0;2;0)$ $A_2 = (0;0 ;0)$ $S = (0;1;3)$ Let $A_4 = (a_1, a_2, a_3)$. What is maximum of $a_1 + a_2 + a_3$ over all $A_4$?
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1answer
25 views

Prove that the parabloas are mutually perpendicular.

Given that two parabolas have the same focus with their axes of symmetry in opposite directions. Then I have to prove that the two intersect at right angles. As I think, because the axes are ...
0
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1answer
15 views

Finding parameters for which the line lies in the plane

I tried to solve the following task: A line L has equation: $\frac{x-2}{p} = \frac{y-q}{2}= z-1$, where $p,q \in \mathbb{R} $. A plane P has equation: $ x +y +3z = 9$. Given that line L lies in the ...
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1answer
28 views

Coordinate Geometry - Parabola [on hold]

Problem: A quadrilateral is inscribed in a parabola, then (A) Quadrilateral may be cyclic. (B) Diagonals of the quadrilateral may be equal. (C) All possible pairs of adjacent side may be ...
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1answer
70 views

Differential equation of Rabbit Fox Problem

A rabbit is hiding at $(0,1)$ and a fox is standing at the point $(0,-1)$ and a big tree is at $(0,0)$ so that rabbit is safe from the attack of Fox. Now the Fox moves in horizontal direction at a ...
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1answer
15 views

Equation of sphere through 2 points.

I require equation of sphere through 2 points that may contain some unknown parameters. I know equation of sphere when two points are the diameter. I tried modifying that but doesn't seem to work. I ...
1
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1answer
26 views

Find the point that is at distance $1$ from $(0,0,0)$ and at distance $3$ from $(1,2,3)$ that is closest to $(5,-2,4)$.

I have this question : Find the point that is at distance $1$ from $(0,0,0)$ and at distance $3$ from $(1,2,3)$ that is closest to $(5,-2,4)$. Here is my failed attempt. I used Lagrange ...
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4answers
73 views

Meaning of $x^2+y^2=0$ (imaginary can have real property?!)

While working for some homework problems for circle to select the radius for cicrles, I encountered with radius 0 and centre at origin i.e., $$x^2+y^2=0$$. When I asked about it to the teacher, he ...
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0answers
24 views

Is there a rational parametrization of Quadric surfaces?

Does there exists a rational parametrization of quadratic surfaces? In particular, I want to parametrize hyperboloid of one sheet $\frac{x^2}{b}+\frac{y^2}{4b}-\frac{z^2}{4b}=1$ where $b$ is rational. ...
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0answers
10 views

Coordinate geometry- Reflection based one (analytical geometry)

The point R is the reflection of the point (−1, 3) in the line 3y + 2x = 33. Find by calculation the coordinates of R
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2answers
58 views

How to determine if any point is in the acute or the obtuse angle between 2 planes

Show that the point $(3, 2, -1)$ lies inside the acute angle formed by the planes $5x-y+z+3=0$ and $4x-3y+2z+5=0$. I have tried this by calculating the angles between the plane, passing through the ...
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0answers
45 views

Given three points in the Cartesian plane, what are the coordinates of the Fermat point?

Given three points $P(x_1,y_1),P(x_2,y_2) \text{ and } P(x_3,y_3)$ in the Cartesian plane, what are the coordinates of the Fermat point? https://en.wikipedia.org/wiki/Fermat_point
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2answers
35 views

Line distance between points (p,q) and (q,p)

I'm trying to find the distance between the points (p,q) and (q,p). As far as I can tell, my steps are correct, but I'm getting ...
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votes
2answers
33 views

Finding the equations of the two lines through $(1,-3)$ tangent to $y=x^2$ [closed]

Two lines through the point $(1,-3)$ are tangent to the curve $y=x^2$. Find the equations of these two lines. Please, any ideas?
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0answers
18 views

Angle between lines whose direction cosines are related by given equations

Find the angle between the lines whose direction cosines l, m & n are linked by the following two equations. $l+m+n=0$ $mn/(q-r)+nl/(r-p)+lm/(p-q)=0$ Where p, q and r are constants. Answer ...
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0answers
26 views

Find the area bounded by the two Ellipses $E_1$ and $E_2$

$E_1$ is an ellipse with center at $(0,0)$ and major axis alone the line $3x+y=0$ with eccentricity $e_1=\frac{\sqrt{3}}{2}$ $E_2$ is another ellipse with center at $(-2.5,10)$ and Major axis alone ...
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2answers
28 views

A simple problem over reflection of point about a line

My question is extremely dull, so please don't bother: I want to have an explicit expression for the reflection of a point $(p,q)$ about a line $y=mx$, in terms of coefficients $p$,$q$ and $m$. But I ...
0
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1answer
21 views

Linear algebra: Analytic geometry problem.

Let $p$ be a line given by the equation $x-1=\frac{y}{2}=\frac{z+3}{2}$, and $q$ a line with the equation $\frac{x}{2}-1=y-2=\frac{z+1}{2}$. If we reflect the $p$ over the plane $\Pi$ we get the line $...
2
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0answers
55 views

Rudin's Proof about Winding Numbers

This is kind of a softball question, an untied loose end that has always bugged me. It is well-known that if $\Gamma_1\sim \Gamma_2$ are two homotopic closed paths in a region $\Omega$, and if $\alpha\...
0
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1answer
54 views

How to plot $(x^2+2xy-24)^2+(2x^2+y^2-33)^2=0$ by hand?

I am trying to solve this problem from Kindle's Analytic Geometry book (Chapter 2, problem 11). I have to plot, by hand, the equation: $(x^2+2xy-24)^2+(2x^2+y^2-33)^2=0$ I can't figure out whether ...
0
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1answer
38 views

Find third triangle vertex given other 2 and lengths, without trigonometry

Given the coorinates of points A, B and lengths of all sides, point C should be found. I have a solution which relies on tangent equation and cosine rule $φ_1 = \arctan2(B_y - A_y, B_x - A_x)$ ...
0
votes
1answer
22 views

Finding coordinate in a quadrilateral

Im trying to do a simulation using Matlab to solve some fluid problem. For this problem I have the following shape: enter image description here For each black point I know the (x,y) coordinates. I ...
1
vote
1answer
68 views

Co-ordinate Geometry : Circle

Problem: Circles $x^2 + y^2 = 1$ and $x^2 + y^2 - 8x + 11 = 0$ cut off equal intercepts on a line through the point $(-2, \frac{1}{2})$. Find the slope of the line. Comments: First I write the family ...
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0answers
24 views

constructing a spiral

picture of spiral]1I am seeking to construct a spiral with the constraints shown in the picture. The problem is that I have not covered polar co-ordinates and I am quite bogged down in understanding ...
2
votes
2answers
50 views

Unit circle shifted upwards so it is tangent the graph of $f(x)=x^{2}$

How can we find k so that $x^{2}+(y-k)^{2}$ is tangent to the graph of $f(x)=x^2$?
3
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1answer
25 views

Can quadric surfaces be made by cutting a 4-dimensional cone?

In my high school multivariable calculus class, we recently learned of quadric surfaces. Since they appeared to be a generalization of conic sections to 3 dimensions, I wondered if they could be ...
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0answers
20 views

Optimum mapping between tesselated parallelograms and tesselated rectangles?

I have a lattice whose points are the vertices of many tessellated parallelograms. Each point is located at $\mathbf{x}=\alpha \mathbf a + \beta \mathbf b$ where $\alpha$ and $\beta$ are integers and $...
0
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1answer
38 views

Prove that locus of vertex is $(a+b)(x^2+y^2)+2h(x\beta + \alpha y) + (a-b)(x\alpha - y\beta)=0$

The base of a triangle passes through a fixed point $(\alpha ,\beta )$. Let the perpendicular bisectors of the sides be the lines $ax^2+2hxy+by^2=0$. It is to prove that the locus of the vertex is : $$...
0
votes
1answer
45 views

Equation of line: find $p + q$

The question: The following two lines intersect, forming an angle of $60°$: $$ \frac{x-1}a = \frac{y-2}{a+1} = \frac{z-1}{a-1} \\ x = y \; \& \; z = 1. $$ If $a = -(p/q)$ where $p$ and $q$ are ...
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vote
1answer
26 views

For pair of st. lines , length of line joining feet of perpendiculars from $(f,g)$ to them is$\sqrt {4.\frac {(h^2-ab)(f^2+g^2)}{(a-b)^2+4h^2}}$

Consider a pair of straight lines through the origin, $$ax^2+2hxy+by^2=0$$ This can be written as, $$y=m_{1,2}x$$ where $m_{1,2}=-\frac {a}{h±\sqrt {h^2-ab}}$. Now, suppose a point $(f,g)$ whence ...
0
votes
1answer
20 views

How to show that the bipolar co-ordinates are othogonal

How to show that the bipolar co-ordinates are othogonal where $x=\dfrac{\sin hv}{\cos hv-\cos u},y=\dfrac{\sin u}{\cos hv-\cos u},z=z$ where $u\in [0,2\pi]$ and $y,z\in (-\infty,\infty)$. ...
0
votes
1answer
21 views

Parametrization of distance to non-unit circle/sphere with non-centered origin

I attempt to parametrize the distance $z(\theta;r,z_0)$ from the origin $(x,y)=(0,0)$ of my coordinate system to arbitrary points $(x,y)$ on a circle, as a function of the variable $\theta$ (angle), ...
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0answers
20 views

Holomorphic bijection from intersection of two circles to a region between two rays

What is a holomorphic map from the nonempty intersection of two circles with "tip points" $a$ (below) and $b$ (neither of which are included in one another) to the region $A$ between two rays? Pictue ...
2
votes
3answers
247 views

Finding the intersection points of a line with a cube

The following is an old high school exercise: Let $A = (5, 4, 6)$ and $B = (1,0,4)$ be two adjacent vertices of a cube in $\mathbb{R}^3$. The vertex $C$ lies in the $xy$-plane. a) Compute the ...
0
votes
1answer
23 views

General Formula for Volume of a Torus with different shaped cross sections

The formula for the volume of a torus is $(\pi r^2) \times 2\pi R$. Does the formula (area of the cross-section) $ \times 2\pi R$ generalize to the volume of all tori? And is there a proof for it (...
5
votes
6answers
79 views

An equilateral triangle is inscribed in a circle of radius $r$. If $P$ is any point on the circumference, find the value of $PA^2 + PB^2 + PC^2$.

An equilateral triangle is inscribed in a circle of radius $r$. If $P$ is any point on the circumference find the value of $PA^2 + PB^2 + PC^2$. I have managed to solve this problem using co-...
2
votes
1answer
59 views

If $p,q,r$ be lengths of perpendiculars from vertices of triangle $ABC$ on any line, prove $a^2(p-q)(p-r)+b^2(q-r)(q-p)+c^2(r-p)(r-q)=4\Delta^2$

Let : $$A:=(x_1,y_1),$$ $$B:=(x_2,y_2),$$ $$C:=(x_3,y_3)$$ be the vertices of the triangle $ABC$. Consider an arbitrary straight line in perpendicular form $x\cos \theta + y\sin \theta - t = 0$. Then ...
0
votes
0answers
30 views

How to find the coordinates of the tangency points on direct common tangent line

I have to find the coordinates of the tangent points on 2 semi-circles that the center of one of them has an offset of x and y from the center of the other one, I need to draw a tangent line between ...