# 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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### Find the area of the region containing the points inside the ellipse $2x^2+y^2-2xy-4x=0$ but outside the ellipse $2x^2+y^2+2xy-4x=0$.

Find the area of the region containing the points inside the ellipse $2x^2+y^2-2xy-4x=0$ but outside the ellipse $2x^2+y^2+2xy-4x=0$. I was able to find the required area using integration. But it is ...
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1 vote
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### What are these conics invariant under linear maps?

Let $$A=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$$ be a matrix with determinant $1$. Then one can see that the conic given by the equation $$Q(x,y)=cx^2+(d-a)xy-by^2=C,\quad C\geq 0$$ is ...
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### Intersection of cones and planes

I need to calculate the volume of the region bound by : The cone $z^2=x^2+y^2$ The plane $z=2x+2y-2$ The plane $z=4$ I have already tried setting up a triple integral but I am having some problems ...
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### There is a compass-like tool that can draw $y=x^2$ on paper. Is there one for $y=x^3$?

Is there a tool that can draw $y=x^3$ on paper? I'm referring to low-tech tools, e.g. not computers. I only know of tools that can draw $y=x^2$. The YouTube video "Conic Sections Compass" ...
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### Condition for angle between two lines less than 90° [closed]

I have asked to find the condition such that the lines joining focii of an ellipse don't subtend right angle at any point on ellipse.. pllz tell me the condition which i have to apply btw the two ...
1 vote
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### Why do the conic sections differ in these two envelopes?

I graphed a square with sides: $y=x$ $y=-x+40$ $y=x+40$ $y=-x$ Then I divided each side to 20 equal parts and drew lines through points: $(1,39)$ & $(19,19)$: (line 1) $(2,38)$ & $(18,18)$: ...
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### True or false: For every $n\in\mathbb{Z^+}$, there exist $a,b,c$ such that $y=(x-a)^2+b$ and $y=c$ enclose exactly $n$ lattice points.

It is easy to show that, for every $n\in\mathbb{Z^+}$, there exists a circle that encloses exactly $n$ lattice points (points with integer coordinates). Can we say the same thing about a parabola and ...
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### How to draw a parabola using basic equipment?

A straight line can be drawn with a straightedge. A circle can be drawn with a compass. An ellipse can be drawn with string and pins. How can we draw a parabola, using basic equipment? Remarks The ...
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### Geometry of underdetermined regression loss function in $n$-dimensions

Let's say I have a regression loss function defined as $(AX-y)^2$, where $A \in \mathbb{R}^{m \times n}$, $X \in \mathbb{R}^{n \times 1}$ and $y \in \mathbb{R}^{m \times 1}$ If $A$ is well determined, ...
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### Let $E$ be an ellipse in the plane. How can we construct an equilateral triangle whose vertices are in E? [closed]

Can an equilateral triangle be constructed from an arbitrary point $A$ on the ellipse such that that the other two vertices are on the curve?
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### Geometric construction of an ellipse enscribed within a in irregular quadrilateral

I am trying to construct the faces of a cube in 3 point perspective, with ellipses enscribed in the same way as shown in this post. I can only use a straight edge and compass, but I can construct an ...
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### Locus of the focus of an ellipse sliding along the coordinate axes? [duplicate]

I approached this problem by taking the centre be points p,q, I know the locus of the centre of the ellipse by the director circle property and added lengths aecos(theta) and aesin(theta) to the ...
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### Three properties of a parabola with three tangents

About ten days ago I discovered three beautiful properties of a parabola with three tangents drawn using the GeoGebra program. I would like to get proof of these properties and also know if any of ...
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### Normalize speed of parametric function

I have a parametric function for an ellipse: $$f\left(t\right)=\left(a\cos\left(t\right),b\sin\left(t\right)\right)$$ As the function goes linearly through t from 0 to 2pi, the point speeds up near ...
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### Angles subtended by tangents from a point on the focus of a conic section

I know that the tangents from a point to a conic section subtend equal angles on the focus. However, I have mostly studied conic sections from the perspective of coordinate geometry, so even when ...
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### Can an ellipse have integer values for perimeter and area?

Is it possible to find an ellipse whose perimeter and area are integer values? I guess that such ellipse doesn't exist, but I haven't fully proved that. Obviously if both the semiaxes are algebraic, ...
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