Questions on conic sections and their properties; the curves formed by the intersection of a plane and a cone. Circles, ellipses, hyperbolas, and parabolas are examples of conic sections.

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Existence of a real empty containing two conjugate pairs.

Given two conjugate pairs of points in general position in $\mathbb{CP}^2$, there is a pencil of real conics containing these four points. Is there a real empty conic in this pencil?
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Finding the area of an equilateral triangle on an ellipse

The question is as follows: Let $E$ be an ellipse with major axis length $4$ and minor axis length $2$. Inscribed an equilateral triangle $ABC$ in $E$ such that $A$ lies on the minor axis and $BC$ is ...
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Can one prove Brianchon's theorem using Ceva's theorem?

Can I prove Brianchon's theorem using Ceva's? I am also wondering if parabola and hyperbola can be inscribed in a hexagon?
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1answer
22 views

Why does the focus point distances of an ellipse sum up to the length of the major axis diameter

Why does the distances from the focus points of an ellipse to arbitrary point in the ellipse sum up to the length of the diameter of the ellipse in the major axis? In other words, how to prove: I ...
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1answer
26 views

Properties of the positive definite Hessian matrix of a convex function

I'm reading about nonlinear programming and I'm having trouble understanding the cool properties that a positive definite Hessian matrix $Q$ of $n$-dimensional function $f: \mathbf{R}^n\rightarrow ...
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4answers
29 views

Find depth of a half-filled parabolic cross-section

Given a cross-section of an object that is parabolic in shape, how do you find the depth of the object when it is "half full". A full example given in an exam: A long trough whose cross-section ...
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2answers
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How to determine the point at a set length along a given function (parabola)?

Given a specific function, a parabola in this instance, I can calculate the length of a segment using integrals to sum infinite right angled triangles hypotenuse lengths. My question is, can I reverse ...
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1answer
25 views

Minimal number of points to define a rotated ellipse?

What is the minimal number of points $N$ to uniquely define the semi-major axis $a$, the semi-minor axis $b$ and the rotation angle $\omega$ of an ellipse whose the center is known/fixed (this is ...
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1answer
19 views

equation for parabola --> Equation for parabolic basin

I have a parabolic basin which i am trying to find the equation for so I can reproduce it. I have taken $3$ points along one line of it to find the equation of the parabola, and I'm wondering if there ...
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1answer
24 views

Equation for the length of a chord parallel to either the minor or major axis in an ellipse

I am looking for a way to compute the length of any chord parallel to the minor (or major) axis of an ellipse. In all cases I know the lengths of both axes, and the distance between the chord and axis ...
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Need a parabolic equation using two points and the slope at those points.

Can someone give me a function to solve any parabolic equation that has two known points with known slopes? Thanks much. Example: Point 1: (x1, y1), slope a Point 2: (x2, y2), slope b
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1answer
17 views

intersection of cone axis with plane

So when the plane intersect the cone, the intersection is a conic. Is (or when is) the axis (of the cone) intersection with the plane the focus of the conic?
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find an ellipsoid given its intersection with axes and knowing the lengths of its principal axes

My question is about ellipsoids. I have an ellipsoid in 3D centered at zero so it has an equation: $x^T U \Sigma^2 U^T x = 1$ I know the lengths of it's principal axes (therefore the $\Sigma$ ...
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1answer
28 views

Help with finding the arithmetic mean of all the radii from the center to the edge of an ellipse?

So far I approached this problem computationally, I decided to take all the radii add them up, by distance formula, then divide by the number of radaii. To make the distribution even, I rotated the ...
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2answers
77 views

How to find the determinant of this matrix

I'd like to find the determinant of following matrix $$ \begin{pmatrix} {x_1}^2 & x_1y_1 & {y_1}^2 & x_1 & y_1 \\ {x_2}^2 & x_2y_2 & ...
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1answer
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How to put these equations in standard form [duplicate]

The first one is a ellipse equation: $x^2+4y^2-2x-16y+1=0$ The second one is an parabola equation: $y^2-16x+2y+1=0$
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1answer
24 views

How to put this conic equation in standard form

I need to put these equations in standard form: $$ x^2+4y^2-2x-16y+1=0 \quad \text{and} \quad y^2-16x+2y+1=0 $$
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1answer
49 views

How to determine if two ellipse have at least one intersection point

All of the question are in sequence and related. 1.Given 2 ellipse with the position x1,y1, x2,y2 and the radius a1,b1, a2,b2, construct an equation to determine if both of them has at least one ...
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44 views

Determine whether the intersection of surfaces is a parabola

Let $C$ be the curve of intersection between the cone $z=\sqrt{x^2 + y^2}$ and the plane $z=1-x$. Is $C$ a parabola? I can see that letting $x=t$ we have parametric equations ...
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1answer
85 views

Why $b^2-4ac$ as determinant?

I am curious why $b^2-4ac$ is used as a determinant of a conic section? Like why this specific expression is chosen, why the value is always greater, lesser or equal to zero for hyperbola, ellipse ...
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1answer
38 views

Central angle of an ellipse

If I have an ellipse centered at the origin and know the length of $a$ and $b$ and was given the length of an arc, how can I find the angle that is between the two radius from the center of the ...
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1answer
59 views

Equation of parabola, tangent at vertex [closed]

Two tangents on a parabola are $x-y=0$ and $x+y=0$. If $(2,3)$ is the focus of the parabola, then find the equation of tangent at the vertex. Thanks. My thoughts: Can't figure out anything :(
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Generalisation of the Ellipse equation

I just wanted to know whether or not the following can be the equation for an ellipse: $$r=\frac{r_0}{1+e\cos\left(\frac{\theta}{\gamma}\right)}$$ Where $\gamma\neq 1$. I ask because whenever I ...
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1answer
21 views

Area of an ellipse proportional to integral of cross-ellipse distances?

I am curious if the area of an ellipse can be shown to be proportional to the integral of all cross-ellipse distances. Before I define cross-ellipse distance, I will give a motivating example from a ...
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2answers
52 views

Find normal to ellipse through arbitrary points

I want to find the normal to ellipse through an arbitrary point. There is an array of points located arround a given ellipse (but not on ellipse curve). What I want to find is the normal of each of ...
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67 views

Find intersections of two ellipses who share one fixed point

Given two ellipses $e_1$ and $e_2$ with $$ e_1 = \{x: \lVert{x - F_1}\rVert + \lVert{x - F_2}\rVert = R \} $$ $$ e_2 = \{ x : \lVert{x - F_1}\rVert + \lVert{x - F_3}\rVert = R \} $$ where $F_1$ is ...
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1answer
32 views

Show that the ellipse and the hyperbola are convex

In Spivak's chapter on differentiation, he asks the reader to prove that the tangent line to an ellipse or a hyperbola intersect the figure at exactly one point. How is this done most elegantly? I ...
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1answer
22 views

how to find the foci, directrix, center of a polar conic section. ($r=\frac{4}{5-4sin\theta} $)

I've been trying to figure this out for a bit and haven't found an answer. the equation is this: $r=\frac{4}{5-4sin\theta} $ I know I need to match this up to a conic graph so I divide top and ...
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2answers
28 views

Is equation for ellipse in polar coordinates correct?

Wikipedia gives the following equation for the conic sections in the polar coordinate system: $r = \frac{l}{1+e\cos\varphi}$. According to the article on conic sections, in case of an ellipse $e = ...
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1answer
38 views

Equation of normal to an ellipse

Show that the equation of the normal at the point $x = a\cos(t)$, $y = b\sin(t)$ of the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$ is $$\frac{2a^2 - b^2}{a}$$ Hi, I am not sure how ...
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1answer
36 views

The image of a conic section under the $z^2$ map

My question in short: In some cases, the image of a conic section under the $z^2$ map is still a conic section. Is there an elegant argument to show that? Let $\Gamma$ be a conic section in the ...
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1answer
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Ellipse cutting orthogonally

If the curves $ax^2+by^2=1$ and $a'x^2+b'y^2=1$ cut orthogonally, then : A)$\displaystyle \frac{1}{b}+\frac{1}{b'}=\frac{1}{a}+\frac{1}{a'}$ B)$\displaystyle ...
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2answers
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Parabola and line proof

Given are three non-zero numbers $a, b, c \in \mathbb{R}$. The parabola with equation $y=ax^2+bx+c$ lies above the line with equation $y=cx$. Prove that the parabola with equation $y=cx^2-bx+a$ lies ...
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The reflective property of ellipses

I have following question to proof: An ellipse is revolved about its major axis to generate an ellipsoid. The inner surface of the ellipsoid is silvered to make a mirror. Show that a ray of light ...
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1answer
36 views

Visualise $3x^2 - 2xy - 10x +3y^2 -2y + 8=0$ after the $x,y$ term has been eliminated (using rotation)

This is a continuation from my previous question. I thought it would be better to start a new one since the old one was answered correctly. The equation in question is: $3x^2 - 2xy - 10x +3y^2 -2y + ...
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43 views

Find a suitable rotation that eliminates the mixed term in the equation: $3x^2 - 2xy - 10x +3y^2 -2y + 8=0$

Find a suitable rotation that eliminates the mixed term in the equation: $3x^2 - 2xy - 10x +3y^2 -2y + 8=0$. Now we want to introduce the new coordinates for $x,y$: $$x = \cos \theta X - \sin ...
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A surprising locus of points

I was playing with the setup for Desargues Theorem in Geogebra today and got a very odd-looking (to me) result. I imagine I could grind through this analytically and get an ugly-looking parametric ...
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Given a pair of conics, construct (synthetically) their shared tangent lines

There are various well-known ways to construct the common tangents to a pair of circles; this is an easy one. I also just learned that we can use Pascal's Theorem to construct a tangent to a conic at ...
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1answer
54 views

Type of this Conic section

I want to determine, to which type the following Conic sections belong to: $$ \begin{align} \textrm{(i)}&\quad-8x^2+12xy-6x+8y^2-18y+8=0\\ \textrm{(ii)}&\quad5x^2-8xy+2x+5y^2+2y+1=0 ...
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2answers
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Hyberbolic and Circular (Trig) Functions: Why no parabolic? [duplicate]

There are circular (trig) functions which determine all the points on a unit circle: and which relate to the area swept out by an angle subtended on the circle. -- These functions can of course be ...
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1answer
16 views

Using the Pin-And-String Method to create parametric equation for an ellipse

Starting with an ellipse, defined with the positions of the foci, and with the major axis length (the semi-major axis is half of the length of the string for the pin and string method), I am trying to ...
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1answer
35 views

Disjoint conic sections?

is there any simple way to figure out whether two conic sections (e.g. two ellipses or an ellipse and a hyperbola) are disjoint or intersect each other? The conic sections are expected to be known ...
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Orientation of rectangle on conic section

Consider a conic section. There are 2 rectangles such that all of the 8 vertices of the 2 rectangles lie on the conic section. Further assume that the 2 rectangles have different orientation (ie. a ...
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Group of Points on an Ellipse

I did some tooling around to find an abelian group operation for the set of points on the ellipse $\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=1$, given by ...
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What are the parameters of a parabola

In the following figure I understand the $bx+c$ part. It is simply the equation of a line. But I don't understand where did $ax^2$ came from? What exactly is it? What does $a$ tell us about a ...
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1answer
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Approximating a curve with a parabole at a given point

I wonder if such a task is possible: we have a curve defined implicitly with: $x^4+y^3-xy-1=0\qquad(1)$ I want to find a parabola in a general form of: $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ ... which ...
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1answer
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How to determine if arbitrary point lies inside or outside a conic

Given the general equation of a conic $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $, is there a way to determine if an arbitrary point $(x_1,y_1)$ lies inside or outside of the conic (ex. parabola or ...
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1answer
44 views

Given a skewed ellipse, how to determine its axis lengths?

I am mentoring a student who is working on a library to import Adobe Pagemaker documents into LibreOffice. Pagemaker represents ellipses as a bounding box (of the original, untransformed ellipse) and ...
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Fascinating Lampshade Geometry

Today, I encountered a rather fascinating problem in a waiting room, which is embodied in the image below. Notice how the light is being cast on the wall? There is a curve that defines the ...
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xy points in the perimeter of a rotated ellipse

How can I calculate the $(x,y)$ position of every point on the perimeter of a rotated ellipse? I have found the equations for a non-rotated ellipse $x=a \cosθ$ $y=b \sinθ$ What are the formulas if ...