# Tagged Questions

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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### 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 ...
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### Why does $\cos(x) + \cos(y) - \cos(x + y) = 0$ look like an ellipse?

The solution set of $\cos(x) + \cos(y) - \cos(x + y) = 0$ looks like an ellipse. Is it actually an ellipse, and if so, is there a way of writing down its equation (without any trig functions)? What ...
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### The Ellipse Problem - finding an ellipse inside a triangle

The problem statement is as follows: A triangle is dissected into six smaller triangles by its angle bisectors. Prove that the intersections of the angle bisectors of each of these smaller triangles ...
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### Calculating Distance of a Point from an Ellipse Border

I'm thinking about using oriented ellipses to represent curves (dents/bumps etc.) in my physics engine, and have a few questions about working with them: What methods are there to finding the ...
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### An interesting property between a hyperbola & parabola

It is well known that when two tangents to a parabola are perpendicular to each other, they intersect on the directrix. In other words, the intersection point of the two tangents make a straight line, ...
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### Are there parabolic and elliptical functions analogous to the circular and hyperbolic functions sin(h),cos(h), and tan(h)?

Are there parabolic and elliptical functions analogous to the circular and hyperbolic functions sin(h),cos(h), and tan(h)? Also, in matters of conic sections, are there other properties such that it ...
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### Calculating a Point that lies on an Ellipse given an Angle

I need to find a point (A on this diagram) given the center point of the ellipse as well as an angle. I've been melting my brain all day (as well as searching through questions here) testing out ...
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### What Does Homogenisation Of An Equation Actually Mean?

For example, if we have a conic; ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 What does homogenising this equation with another line (say ax + by + c = 0 ) actually mean? As in, what are the graphical ...
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### Finding Maximum Area of a Rectangle in an Ellipse [duplicate]

Question: A rectangle and an ellipse are both centred at $(0,0)$. The vertices of the rectangle are concurrent with the ellipse as shown Prove that the maximum possible area of the ...
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### Need to find the ellipse of maximum area inscribed in a semicircle.

An ellipse inscribed in a fixed semi circle touches the semi-circular arc at two distinct points and also touches the bounding diameter. Its major axis is parallel to the bounding diameter. ...
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### 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 ...
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### Prove that $|PF_{1}|+|PF_{2}|$ is Constant in an Elipse

Given an elipse with two focus $F_{1}$ an $F_{2}$, and $A$ is an arbitrary point at the elipse. Stright line $AF_{1}$ has another intersection point $B$ with the elipse, and $AF_{2}$ has another ...
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### Generalization of ellipse equation to higher dimensional surfaces

This question was motivated by Definition of an ellipsoid based on its focal points . I'd like to avoid terms like ellipsoid, so I'll use terms like one-dimensional ellipse (a normal ellipse in the ...
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### How to find an ellipse , given 2 passing points and the tangents at them?

Please answer to a question , how to find an ellipse which passes the 2 given points and has the given tangents at them. And one related question is that the given condition can decide just one ...
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### Intersection of conics using matrix representation

I came across a very interesting section of a wikipedia article on conics: http://en.wikipedia.org/wiki/Conic_section#Intersecting_two_conics I am trying to work out a couple of examples to add to ...
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### Parabolas through three points

We can draw an infinite number of parabolas that pass through three given points $A$, $B$, $C$ (in that order). For each such parabola, we take the tangent lines at $A$ and $C$, and intersect them to ...
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### How to geometrically prove the focal property of ellipse?

How to prove geometrically that if we have a tangent of ellipse with focus F and F' in point P, that tangent is bisector of the angle between a line joining focus F to point P and the line F'P outside ...
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### Finding upper segments of intersecting parabolas

I have multiple parabolas ($y = ax^2 + bx + c$) which may intersect with each other (or some of them may not intersect). I am trying to find upper segments of these parabolas, e.g. bold part in the ...
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### 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 ...
There is a natural bijection between conics (written as homogeneous quadratic) and 3x3 matrices: C=aX^2+2bXY+cY^2+2dXZ+2eYZ+fZ^2\Leftrightarrow \left(\begin{array}{ccc} a&b&d\\ b&c&...