Questions tagged [conic-sections]
For questions about circles, ellipses, hyperbolas, and parabolas. These curves are the result of intersecting a cone with a plane.
5,230 questions
2 votes
1 answer
80 views
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 ...
- 357
6 votes
1 answer
154 views
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 $\epsilon_C$...
- 63
1 vote
1 answer
65 views
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 ...
user1709783
2 votes
1 answer
47 views
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 ...
- 47.2k
7 votes
1 answer
220 views
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 ...
- 4,451
1 vote
0 answers
25 views
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 ...
- 695
0 votes
2 answers
104 views
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 $90^\circ$. Let $T$ be a point such ...
- 121
4 votes
2 answers
118 views
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 ...
- 695