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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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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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,ドル $B$ and $C,ドル and the three ellipses $\epsilon_A,ドル $\epsilon_B$ and $\epsilon_C$...
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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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user1709783
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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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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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Prescribing 5ドル$ 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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Trace of intersection of two perpendiculars is hyperbola
The problem:
Let $F$ be a point on the positive x-axis. Let $M_1, M_2$ be distinct points on the y-axis such that $\angle M_1 F M_2$ is constant and bigger than 90ドル^\circ$. Let $T$ be a point such ...
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How to obtain a nondegenerate configuration for real parabolas?
Let $P_i=(x_i,y_i)$ be eight distinct points in the plane, expressed in Cartesian coordinates.
Define
$$
m_{ij}=\frac{y_i-y_j}{x_i-x_j}.
$$
A quadruple of points $(P_i,P_j,P_k,P_\ell)$ is said to be ...