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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The minimal value of a fraction based on a focal chord of an ellipse [duplicate]

I came across a very interesting olympiad problem. It goes as follows: suppose you have an ellipse given by $\frac{x^2}{16}+\frac{y^2}{9}=1$ and a line that goes through the point $A(\sqrt{7},0)$. ...
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On a closure theorem involving 3 conics and an inscribed 3$n$-gon whose sides pass through 3 fixed points

(Note: I'm not a native English speaker.) When I was playing around with Geogebra, I personally discovered some interesting properties concerning conic sections. Here is one of them. Consider the ...
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Rotation matrix to construct canonical form of a conic

I want to find the canonical form of the following conic: $$C: 9x^2+4xy+6y^2-10=0.$$ I've found $C$ is a non-degenerate ellipses (computing the cubic and the quadratic invariant), and then I've ...
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Algorithm to determine if a 3D ellipsoid is contained within another?

Can anyone point me to an algorithm for how to efficiently check if a 3D ellipsoid is contained within another one? We can assume their origins are collocated. I am dealing with covariance ellipsoids ...
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Find domain and range of the slanted hyperbola

Given the conic section $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ and I know that it is a hyperbola and $B\ne 0$. How to find its domain and range? I guess the method of Lagrange multipliers will fail here.
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Construct focii of ellipse given center and four tangent lines

We are given 4 distinct lines $a,b,c$ and $d$ which are said to be tangents to an ellipse. Let's consider that the 4 meetings of these lines form an convex quadrilateral $ABCD.$ There is a theorem ...
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Angle between normal vector of ellipse and the major-axis.

I am trying to derive the angle made between the major or x-axis and the normal vector of an ellipse of general shape $x = a\cos(t),y=b\sin(t)$ with the parameter $t$ reffering to Ellipse in polar ...
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Find a point $P_2$ on an ellipse, whose chord with $P_1$ is a max distance $d$ from its nearest side

I'm not sure if this solution is available in closed form, but after drawing it out I do think there will be two unique solutions always. I unfortunately have no clue where to start. Given: An ...
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find the conic curve through given points

I am trying to find a conic curve that passes through $[0:1:2]$ and intersects the conic curve $F =x_1^2 + 4x_1x_2 + 4x_0x_2 -x_0^2$ at the point $[1:1:0]$ with multiplicity $4$. Since I am given $5$ ...
Is an equation only a parabola if it is quadratic? Could the graph of $y = x^{1.65}$ be described as parabolic in shape?
If given the equation $y = x^{1.65}$, could it be described as parabolic in shape, or does the equation have to have $x^2$ as its highest degree term?
What is the difference between hyperbola and ellipse, since they both have the equation, $\frac{y^2}{a^2}+\frac{x^2}{a^2-c^2}=1$?
Noted that the equation of ellipse is given by $\dfrac{y^2}{a^2}+\dfrac{x^2}{b^2}=1$,where $b^2=a^2-c^2$ While the equation of hyperbola is $\dfrac{y^2}{a^2}-\dfrac{x^2}{b^2}=1$,where $b^2=c^2-a^2$ ...