For questions about circles, ellipses, hyperbolas, and parabolas. These curves are the result of intersecting a cone with a plane.
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When an ellipse touches the sides of a triangle
An ellipse touches the sides of a triangle $abc$ from inside in the points $a',b',c'$. How can I prove, that the lines $ aa',bb',cc'$ meet in one point? The ellipse equation is : $ \frac{x^2}{A^2} + \... geometry conic-sections- 131
Approximating an Ellipse given 4 points.
I am facing a problem in which I need to create masks for a certain region. This region is not a perfect ellipse, but for all intents and purpose, the ellipse needs to encapsulate these 4 coordinates. ... conic-sections matlab learning- 31
An unexpectedly difficult geometry problem
I've run into a geometry problem that feels like it should have an easy answer. But short of numerical integration, I can't find a way to solve it. Consider a filled circle on top of a filled ellipse ... integration geometry circles conic-sections area- 169
How exactly do I 'show' that an equation represents something?
Here is an equation and I want to know how I can show that this represents an ellipse ($a$ and $b$ are complex constants and $\alpha$ is a real variable): $$ z = ae^{i\alpha} + be^{-i\alpha} $$ I ... complex-analysis complex-numbers conic-sections complex-geometry- 13
Does the line through the center of a right cone always pass through one of the foci of each ellipse made with that cone?
Is the line passing through the center of a right cone always one of the foci of such an ellipse made through the intersection of the right cone? (Image credit: Courtney Seligman, cseligman.com) conic-sections- 1
Prove that the intersection between a quadric and a plane is always a second-order curve
My problem is simply: If you have a general quadratic surface $$ Q: ax^2+by^2+cz^2+2fyz+2gzx+2hxy+2px+2qy+2rz+d=0.$$ prove that the intersection curve between this suface and a plane $$ P: a_1 x + a_2 ... analytic-geometry quadratics conic-sections surfaces- 41
if a curve $A=\{z : |z-3|+|z+3|=8\}$ and $B=\{z : |z-3|=k\},\; k\in \mathbb R^+$ touch the curve $A$ internally, then which are correct
if a curve $A=\{Z : |Z-3|+|Z+3|=8\}$ and $B=\{Z : |Z-3|=k\},\; k\in \mathbb R^+$ touches the curve $A$ internally, and given another curve $C = \bigl\{Z : \bigl||Z-3|-|Z+3|\bigr|=4\bigr\},\;$ where $Z$... algebra-precalculus conic-sections hyperbolic-geometry- 1,996
How to find the angle of an ellipse from the origin?
I have the following problem where I'm trying to place the center of the smaller black ellipse where the red diagonal line crosses the blue ellipse (where the arrow is pointing), but I've only been ... trigonometry conic-sections- 187
show that $\tan \alpha_i = -t_i$ for each $i=1,2,3$
My question pertains to the question and solution below. In particular, can someone prove why $\tan \alpha_i = -t_i$ for each $i$? I think it would be worth clarifying what the angle each normal makes ... geometry trigonometry contest-math conic-sections- 1,109
How to find the equation of a Hyperbola given two foci and a vertex [closed]
If I know the coordinates of the foci F1, F2 and the coordinate of a vertex P1 that lies on the hyperbola (both expressed in 2D cartesian coordinates). How would I determine the equation of the ... geometry analytic-geometry conic-sections- 147
tangents and asymptotes to hyperbola
I seem to have a lot of confusion in understanding the concept of asymptotes to a curve, particularly to hyperbola(it seems very difficult for me to visualize since there are two branches to a ... asymptotics conic-sections curves tangent-line- 21
Analytical solution for ratio of circle to ellipse area
I am interested in finding an analytical solution for the ratio of the area of a circle within an ellipse to the area of an ellipse. so the ratio of the magenta area to that of the magenta + cyan area.... geometry circles conic-sections area- 115
Symmetries of multifocal ellipses
A classical ellipse has two foci (I hope my English is correct saying this) and admits the Klein group as isometry group, whose elements are the identity, the reflexions along the small and larges ... permutations euclidean-geometry conic-sections symmetry- 2,486
Perimeter of Ellipse $x^2+\frac{y^2}{2(\sqrt{2}-1)}=1$
Prove that the perimeter of ellipse $x^2+\frac{y^2}{2(\sqrt{2}-1)}=1$ is $P=\frac{\Gamma^2(\frac{1}{8})}{2^\frac{7}{4}\Gamma(\frac{1}{4})}+\frac{2^\frac{11}{4}.\pi.\Gamma(\frac{1}{4})}{\Gamma^2(\frac{... geometry functions conic-sections- 1
main axis transformation with the conic $Γ:−6x^2−6yx+4x+2y^2−4y+1=0$
The purpose of this exercise is to reduce the conic $Γ:−6x^2−6yx+4x+2y^2−4y+1=0$ to the canonical expression. What I already have is the Eigenvectors $-7,3$ and the eigenvectors $\begin{pmatrix} 3 \... linear-algebra conic-sections- 323
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