Questions on conic sections and their properties; the curves formed by the intersection of a plane and a cone. Circles, ellipses, hyperbolas, and parabolas are examples of conic sections.

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65
votes
2answers
731 views

Modelling the “Moving Sofa”

I believe that many of you know about the moving sofa problem; if not you can find the description of the problem here. In this question I am going to rotate the L shaped hall instead of moving a ...
0
votes
1answer
12 views

Find the minimum distance to move an ellipse to be inside another ellipse?

For the problem of ellipse intersection, I would like to know an accurate "general, including the case of two non intersected ellipses" method to calculate the minimum ecludian distance to translate ...
0
votes
1answer
19 views

Discriminant of equations of conic sections [on hold]

How to find discriminant of the equation of conic sections. Like discriminant of $ax^2+2hxy+by^2+2gx+2fy+c=0$ is $abc+2fgh-af^2-bg^2-ch^2$. Is there any special formula to calculate this.
9
votes
6answers
607 views

538.com's Puzzle of the Overflowing Martini Glass - How to compute the minor and major axis of an elliptical cross-section of a cone

FiveThirtyEight.com Riddler Puzzle / May 13 The puzzle goes like this; "It’s Friday. You’ve kicked your feet up and have drunk enough of your martini that, when the conical glass (🍸) is upright, the ...
1
vote
1answer
1k views

find a circle tangent to an ellipse

As shown in the figure, the circle is moving upwards along the line $x=x_0$ suppose we know the following parameters: $a,b,x_0,r$ The ellipse equation is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$; The ...
2
votes
2answers
96 views

Minimum Area of An Ellipse Surrounding Four Circles

The circles are all four combinations of $(x\pm60)^2+(y\pm25)^2=5^2$ (see pic at end). The ellipse I've got is one I found via trial and error but there must be an analytical way to solve this, ...
3
votes
3answers
72 views

How to show that any rectangle in ellipse must be oriented parallel to axes?

A problem which is often given as an exercise for students learning about calculus and finding extrema, is to find maximal possible area of a rectangle inside an ellipse. Such question was asked, for ...
2
votes
3answers
91 views

4-ellipse with distance R from four foci

I'm trying to find the equation for the generalization of an ellipse called a $n$-ellipse which has a constant distance R from four foci located at $(0,0),(0,1),(1,0),(1,1)$ Edit: As an algebraic ...
3
votes
1answer
441 views

If $x^2 + y^2 + Ax + By + C = 0 $. Find the condition on $A, B$ and $C$ such that this represents the equation of a circle.

If $x^2 + y^2 + Ax + By + C = 0 $. Find the condition on $A, B$ and $C$ such that this represents the equation of a circle. Also find the center and radius of the circle. Here's my solution, ...
9
votes
1answer
267 views

Right triangle on an ellipse, find the area

Beginning note: Please wait until the animations load. The loading might take some time depending on your internet connection. Secondly, the title and the content of the question might not be well ...
0
votes
1answer
21 views

Point of intersection of ellipses

If two ellipses are intersecting at a point,is it necessary that the line drawn joining the centre of those two ellipses should also pass through the point of intersection (of ellipse)? (if yes,how to ...
-1
votes
0answers
28 views

Condition that a line touches a conic [on hold]

What is the condition that the line $k/r=A + B\cos\theta$ touches the conic $k/r = 1 + e\cosθ$? 
1
vote
2answers
487 views

Is it possible to calculate the volume of a parabolic arch?

Given that you know the equation of a parabola that only has positive values, is it possible to find the volume of the parabolic arch itself? NOT the volume of space underneath the arch. I asking ...
1
vote
1answer
20 views

Calculator limits on a parabola

Hi guys I'm making Patrick Star for a graphing project. Anyways I'm using a parabola for his head on my TI-84 but when I set limits on it, it graphs a straight line. So the equation itself is ...
1
vote
2answers
43 views

Conic Sections: Hyperbola (Finding the Locus)

This is a multipart question so bear with me until I get to the part where I am stuck on. $H$: $xy=c^2$ is a hyperbola (i) Show that $H$ can be represented by the parametric ...
0
votes
1answer
672 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
2
votes
2answers
341 views

Area of Parallelogram in an Ellipse

A parallelogram is inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with the fixed line $y=mx$ as one of its diagonals. Prove that the maximum area of the parallelogram is $2ab$. ...
2
votes
4answers
1k views

Intersection of ellipse with circle

I would like know whether a circle is intersecting an ellipse. Here ellipse equation is $$Ax^2 + Bxy + Cy^2 + dx+ey + 1 = 0,$$ and the circle equation is $$(x-g)^2 + (y-f)^2= r^2.$$
3
votes
2answers
652 views

Ellipse bounding rectangle

I'm trying to find the ellipse that bounds a rectangle in a way that the "distance" between the rectangle and the ellipse is the same vertically and horizontally. Here is an image to illustrate what ...
6
votes
5answers
17k views

Find the area of largest rectangle that can be inscribed in an ellipse

The actual problem reads: Find the area of the largest rectangle that can be inscribed in the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.$$ I got as far as coming up with the equation ...
1
vote
1answer
2k views

How to compute the chord length of an ellipse?

How do I calculate the chord length of ellipse? I need to design a blade vane rotating inside an elliptical profile. The blade is supposed to be in contact with ellipse with a maximum clearance of ...
23
votes
4answers
25k views

Check if a point is within an ellipse

I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane. How do I determine if a point $(x,y)$ is within the area bounded by ...
0
votes
0answers
28 views

Rotate an Ellipse

$x = h + a \cos(φ) \cos(θ) + b \sin(φ) \sin(θ)$ $y = k + b \cos(φ) \sin(θ) - a \sin(φ) \cos(θ)$ Hi, I have basic question of parametric equation for ellipse. I'm trying to rotate horizontal ellipse ...
0
votes
0answers
94 views

ellipse and segment intersection

I have a rotated ellipse, not centered at the origin, defined by $x,y,a,b$ and angle. Then I have a segment defined by two points $x_1$, $y_1$ and $x_2$, $y_2$. Is there a quick way to find the ...
0
votes
0answers
37 views

Finding $x^2$ and $y^2$ of hyperbola

Currently, I am trying to the $x^2$ and $y^2$ of a hyperbola. I have the vertices at $(-1, -1)$ $(5, -1)$ I have the focus at $(-4, -1)$ $(8, -1)$ I know that the distance between two vertices ...
2
votes
1answer
501 views

Computing a matrix to convert an (x,y) point on an ellipse to a circle

I have an ellipse defined by its semi-major axis, inclination, and position angle. The ellipse is centered on the origin. I would like to solve for a matrix that converts this ellipse to a circle. ...
5
votes
3answers
275 views

How to find center of a conic section from the equation?

If we are given a curve in the form $$ax^2+2bxy+cy^2+2dx+2ey+f=0$$ and the following determinant $$\delta=\begin{vmatrix}a&b\\b&c\end{vmatrix}=ac-b^2$$ is non-zero, then this is either a curve ...
1
vote
2answers
172 views

Prerequisites for Appolonius Conics?

I want to get Thomas Heath's version of Apollonius's Conic Sections. Does anyone know the prerequisites to understand everything in this book? I heard I would need the Euclid's Elements book on Solid ...
6
votes
0answers
76 views

Why does the Ellipsograph/Trammel of Archimedes draw an ellipse, really?

Here's a diagram of the device I mean, hard at work drawing an ellipse. I find this quite surprising, and would like to get to the bottom of things. Essentially, a rod (black line in animation) is ...
0
votes
2answers
155 views

Why are degenerate conics not projectively equivalent to nondegenerate conics?

This is what I understand about conics being projectively equivalent. Two conics $C1=V(F)$ and $C2=V(G)$ are projectively equivalent if there is an invertible matrix $A$ such that $F(X,Y,Z)=0$ iff ...
1
vote
3answers
29 views

Algorithm: Intersection of two conics

I am looking for a detailed description of an algorithm for the classical problem of computing the intersection of two conic curves. The curves are given by two equations of the form: $$ a x^2 + b ...
1
vote
3answers
41 views

Same perimeter and area for a circle and an ellipse

For a given circle, is there exist an ellipse with same perimeter and area as to that circle? If not, that is my suspicion, is in three-dimension parallel question: For a given sphere, is there ...
1
vote
1answer
41 views

Identification of a conic section

Consider the equation $(E)\hskip 5mm x^2+xy+ky^2+6x+10=0$. I am looking for conditions on $k$ for the graph of $(E)$ to be a circle or an ellipse. Clearly, if it is a circle or an ellipse, its ...
0
votes
3answers
34 views

Equation of parabola given 2 points $(x_1,y_1)$ and $(x_2,y_2)$ in expanded form

I need to find an equation for the parabola that passes through the points $(0,0)$ and $(5,0)$, such that $f(x)<0$ whenever $0< x <5$. The answer should be in expanded form. I.e., ...
1
vote
1answer
988 views

Radius vs Radius of curvature of an ellipse

I am a bit confused by the physical meaning of radius vs radius of curvature, with regards to an ellipse. For a standard ellipse: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ In this case, the $a$ ...
1
vote
2answers
45 views

Points $P_i$ on an ellipse such that angle $P_iOP_{i+1}=\frac{\pi}{n}$

Consider an ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ with $O$ as the origin. $n$ points denoted as $P_1,P_2,\cdots$ are taken on the ellipse such that angle $P_iOP_{i+1}=\frac{\pi}{n}$ where ...
5
votes
4answers
204 views

Find the latus rectum of the Parabola

Let $y=3x-8$ be the equation of tangent at the point $(7,13)$ lying on a parabola, whose focus is at $(-1,-1)$. Evaluate the length of the latus rectum of the parabola. I got this question in ...
-1
votes
1answer
69 views

Find the equation of the ellipse

An ellipse with centre at $(4,3)$ touches $x$-axis at $(0,0)$. If the slope of the major axis of ellipse is 1, then find the equation of the ellipse?
0
votes
1answer
35 views

How do parametric equations work?

I was given a graph like this in my exam. Its defined para-metrically by x=c^2 and y =c^3. It won't help me now but could someone explain this to me why I have two seemingly different lines I know ...
2
votes
2answers
2k views

Finding the eccentricity/focus/directrix of ellipses and hyperbolas under some rotation

Suppose I have an ellipse/hyperbola rotated about the origin by some angle $\theta$. Am I right in saying that the following general process will find the eccentricity $e$ of these conics? Find ...
1
vote
1answer
18 views

Focus of a parabola

If (2,0) is the vertex and y-axis the directrix of a parabola find the focus of the parabola. What does y-axis is directrix mean here?
2
votes
2answers
29 views

Slope of axes of a General Conic Section

A General Conic Section is given by the equation $ax^2 + by^2 + 2hxy +2gx +2fy + c =0 $. Let the $\theta$ be the slope of one of its axes. Prove that : $$\tan 2\theta = ...
0
votes
1answer
21 views

Hyperbolas and Quadrants on Rotation

Let's assume we have a standard hyperbola. On rotating the hyperbola $45^{\circ}$ clockwise, the new hyperbola should lie in the $2$nd and $4$th quadrant. However, the equation of a parabola rotated ...
13
votes
8answers
402 views

How do I transpose an ellipse function to stretch the ellipse into curved space?

I'm working on an engineering project, using CAD software. I can write simple parametric functions to draw an ellipse, with $\theta$ ranging from $0$ to $2\pi$ ...
5
votes
2answers
44 views

Why are there only two tangents to a hyperbola from a point, instead of four?

Why are only two tangents possible to a hyperbola from a point? If we treat the hyperbola as two individual parabolas, then a point should be able to create two tangents through it for both of them, ...
0
votes
0answers
13 views

ratio of areas of a cone and plane intersection after the plane is rotated

I have a right cone with the tip $V$ and tip angle $\alpha$, with the axis $d_1$ that it is orthogonal on the plane $P$ at distance $L$ from $V$. In the plane $P$ I have a line $d_2$ that intersects ...
0
votes
2answers
65 views

How to get the properties of an ellipse with six points given.

I am looking for a way to calculate the lengths of both semi-axes and the rotation angle of the ellipse in the image as shown in this picture. Six points are given, with two pairs of points being ...
2
votes
1answer
32 views

Prove that the group of the rational points on the conic $u^2-Av^2=1$ is not finitely generated.

This is an exercise from Rational Points on Elliptic Curves by Silverman. Let $H$ be the conic $u^2-Av^2=1$ where $\sqrt{A}\notin \mathbb{Q}$. If $(u_1,v_1), (u_2,v_2)$ are two points in ...
1
vote
1answer
22 views

Finding Coordinate along Ellipse Perimeter with Arbitrary Origin Coordinates

This is heavily related to: This Question I know that question should have handed me the answer, but I can't quite wrap my head around what I need to do to get coordinates with an arbitrary origin. ...
0
votes
1answer
38 views

circle cuts three circles at the extremities of the diameter

If the circle $$x^2 + y^2 + 2gx + 2fy + c = 0$$ cuts the three circles $$x^2 + y^2 – 5 = 0\space;\space x^2 + y^2 – 8x – 6y + 10 = 0 \space;\space x^2 + y^2 – 4x + 2y – 2 = 0;$$ at the ...