Questions tagged [conic-sections]

For questions about circles, ellipses, hyperbolas, and parabolas. These curves are the result of intersecting a cone with a plane.

A conic section is a smooth planar curve that is the result of intersecting a cone with a plane. There are commonly four conic sections: circles, ellipses, parabolas, and hyperbolas.

We can construct these conic sections analytically. The solutions to the equation $x^2+y^2=z^2$ give us a cone in three-dimensional space. An plane in three-dimensional space that goes through a point $p=(x_0,y_0,z_0)$ and has normal vector $\langle a,b,c \rangle$ is given by the equation


Finding the common solutions to the equation of the cone and equation of the plane for various choices of $p$ and normal vector will—after a change in coordinates to write them as planar curves—lead you to the (potentially) familiar equations

  • Circle: $x^2+y^2 = r^2$
  • Ellipse: $ax^2 + by^2 = r^2$
  • Parabola: $ax^2 +by = r^2$
  • Hyperbola: $ax^2 - by^2 = r^2$

There are also geometric constructions of the conic sections. For example, a circle is the set of all points that are a fixed distance from a given points. An ellipse is the set of all points .... The construction of Dandelin spheres (see Wikipedia) unifies the analytic and geometric constructions of conic sections.

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Modelling the "Moving Sofa"

I believe that many of you know about the moving sofa problem; if not you can find the description of the problem here. In this question I am going to rotate the L shaped hall instead of moving a sofa around the corner. By rotating the hall…
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Do "Parabolic Trigonometric Functions" exist?

The parametric equation $$\begin{align*} x(t) &= \cos t\\ y(t) &= \sin t \end{align*}$$ traces the unit circle centered at the origin ($x^2+y^2=1$). Similarly, $$\begin{align*} x(t) &= \cosh t\\ y(t) &= \sinh t \end{align*}$$ draws the right part…
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Check if a point is within an ellipse

I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane. How do I determine if a point $(x,y)$ is within the area bounded by the ellipse?
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Why does $\cos(x) + \cos(y) - \cos(x + y) = 0$ look like an ellipse?

The solution set of $\cos(x) + \cos(y) - \cos(x + y) = 0$ looks like an ellipse. Is it actually an ellipse, and if so, is there a way of writing down its equation (without any trig functions)? What motivates this is the following example. The…
Mr Bingley
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Using Gröbner bases for solving polynomial equations

In my attempts to understand just how computer algebra systems "do things", I tried to dig around a bit on Gröbner bases, which are described almost everywhere as "a generalization of the Euclidean algorithm and Gaussian elimination". I've tried to…
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What is the general equation of the ellipse that is not in the origin and rotated by an angle?

I have the equation not in the center, i.e. $$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1.$$ But what will be the equation once it is rotated?
andikat dennis
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Parabola is an ellipse, but with one focal point at infinity

While I was reading about conic sections, I came across the following statement: A parabola is an ellipse, but with one focal point at infinity. But it is not clear to me. Can someone explain it clearly?
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Lampshade Geometry Problem

Today, I encountered a rather interesting problem in a waiting room: $\qquad \qquad \qquad \qquad$ Notice how the light is being cast on the wall? There is a curve that defines the boundary between light and shadow. In my response below, I will…
Kaj Hansen
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What do cones have to do with quadratics? Why is $2$ special?

I've always been nagged about the two extremely non-obviously related definitions of conic sections (i.e. it seems so mysterious/magical that somehow slices of a cone are related to degree 2 equations in 2 variables). Recently I came across the…
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Why are certain PDE called "elliptic", "hyperbolic", or "parabolic"?

Why are the Partial Differential Equations so named? i.e, elliptical, hyperbolic, and parabolic. I do know the condition at which a general second order partial differential equation becomes these, but I don't understand why they are so named? Does…
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Looking for a function that approximates a parabola

I have a shape that is defined by a parabola in a certain range, and a horizontal line outside of that range (see red in figure). I am looking for a single differentiable, without absolute values, non-piecewise, and continuous function that can…
14 answers

Derivation of the formula for the vertex of a parabola

I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form $$y = a x^2 + b x + c$$ My teacher gave me the formula: $$x = -\frac{b}{2a}$$ as the $x$ coordinate of the vertex. I asked her why, and…
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How to determine the arc length of ellipse?

I want to determine the length of an arc from the ellipse in the picture below: How can I determine the length of $d$?
Mohammad Fakhrey
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Is this an ellipse?

Is this parameterisation an ellipse: \begin{align}x(t) &= \frac{2 \cos(t)}{1 + a \sin(t)}\\ y(t) &= \frac{2 \sin(t)}{1 + a \sin(t)}\end{align} where $a$ is a real positive parameter. I tried to do it the naive way but couldn't find a…
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The locus of the intersection point of two perpendicular tangents to a given ellipse

For a given ellipse, find the locus of all points P for which the two tangents are perpendicular. I have a trigonometric proof that the locus is a circle, but I'd like a pure (synthetic) geometry proof.
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