Questions tagged [conic-sections]
For questions about circles, ellipses, hyperbolas, and parabolas. These curves are the result of intersecting a cone with a plane.
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A straightforward proof of a property of lines enveloping an ellipse using complex numbers : looking for connections
This question finds its origin in the very fruitful exchange I have had with @Li Kwok Keung, in the framework of this question.
Let us consider the following plane geometrical configuration that I ...
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Proving that $x := x - yt$; $y := y + xt$ traces an ellipse
I was investigating the iterated function formed by the following JavaScript code, trying to find an invariant that can prove that the system doesn't slowly diverge:
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Find out the distance between centers of two intersecting semi-ellipses $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.
There are two identical semi-ellipses, one with center at the origin $O$, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, and the other at $R$, $\frac{(x-d)^2}{a^2}+\frac{y^2}{b^2}=1$.
Find out the distance $d$ ...
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Loci of sliding ellipse foci [duplicate]
An ellipse of major axis and eccentricity $(2a,e) $ slides up and down contacting the coordinate axes $ (x,y)$ always.
What are the loci of individual foci?
At any instant the variable pentagon has ...
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Angle between tangents of a hyperbola
Let a hyperbola with semi major axis length $a$ and shortest radius $r_p$ be given. For $r\geq r_p$ find angle $\gamma$ between the tangent at distance $r_p$ and the tangent at distance $r$ from the ...
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Do aligned midpoints of all parallel chords characterize conics?
A well-known property of conics states that the midpoints of parallel chords lie on a line passing through the center.
Let $K \subset \mathbb{R}^2$ be a strictly convex set with nonempty interior, and ...
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Is this statement about intersecting ellipses a known theorem?
I have recently found a (in my opinion) neat little geometric fact and a proof thereof:
Theorem:
Given three points $A$, $B$ and $C$, and the three ellipses $\epsilon_A$, $\epsilon_B$ and $\epsilon_C$...
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Families of real ellipses with two fixed real and two fixed imaginary tangents: is the common interior al nonempty subset of a line??
Set-up. Work over $\mathbb R^2$. Let $\mathcal F=\{E_t\}$ be a 1-parameter family of real ellipses such that all members have the same four tangents: two real lines and a conjugate pair of complex (...
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Ellipse inscribed in a convex quadrilateral
I am considering the problem of determining the ellipse that is inscribed in a given convex quadrilateral, which in addition has a certain orientation of its axes.
It is known that there is an ...
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Extending to more general conics a property established for chords of circles
This is a follow up of this recent question, now closed.
In order to gather here all the information, let me first recall the question :
Initial (synthetized) question $(Q)$: Being given a circle $(C)$...
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Normals at three parabolic points P,Q,R on $y^2=4ax$ meet on a point on the line $y=k,$ then prove that sides of $\Delta$PQR touch $x^2=2ky$
Normal at a point on the parabola $y^2=4ax$ is given as
$$y=mx-am^3-2am,$$ if normals at three points meet at a point $(x_1,k)$ on the line $y=k$
then we have: $$k=mx_1-am^3-2am \tag{1}.$$
This can ...
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Angle at base of ellipsoid cap for fitting data
Here is the cross-section of an ellipsoid that has rotational symmetry around $b$. It approximates a pinned droplet on a smooth surface (pinned meaning that its contact area is constant while the ...
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A geometric property involving a cyclic quadrilateral and a conic
Yesterday, while experimenting with GeoGebra, I discovered what seems to be a remarkable geometric property involving a cyclic quadrilateral and conic sections. However, I have not been able to prove ...
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Maximize eccentricity over all conics tangent to four fixed non-parallel lines
First, let's agree on the eccentricity of degenerate conics:
The animated gif shows Ellipses, hyperbolas with all possible eccentricities from zero to infinity and a parabola on one cubic surface. ...
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Prescribing $5$ normal lines to a conic: is there always at least one real solution?
Question.
Fix five real lines $\ell_1,\dots,\ell_5$ in the Euclidean plane in general position.
A real conic is a real plane quadratic curve (nondegenerate) in an affine chart.
I would like to show ...
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