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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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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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
79 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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35 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
51 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
20 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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1answer
38 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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61 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
30 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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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
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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
37 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
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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
17 views

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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32 views

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
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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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1answer
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
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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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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
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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
33 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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2answers
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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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60 views

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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Solving the eccentricity of an ellipse with three constants.

I have a question about how to solve the eccentricity of an ellipse when it has three constants. For instance. Imagine a problem has a2/4 squared + b2/ 5 squared + c2/8 squared = 1 and for this ...
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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
83 views

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
37 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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3answers
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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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1answer
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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 ...
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1answer
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maximum radius of a circle inscribed in an ellipse

Consider an ellipse with major and minor axes of length 10 and 8 resp. The radius of the largest circle that can be inscribed in this ellipse, given that the centre of this circle is one of the focus ...
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102 views

How to plot the graph of parabola?

I am new in the conics so there are two confusions: 1. How to plot the graph of this parabola equation $y^2-x-2y-1=0$? 2. what would be the equation of parabola when vertex $(3,2)$ and ends of focal ...
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Practise Calc Question

How is the radius $10$ in the circle equation: $x^2+y^2+6x-4y+3=0$? My work: Standard Form for circle: $(x-a)^2+(y-b)^2=r^2$ $X:$ $X^2+6x+[?]=-3+[?]$ $(x+3)^2= X^2+6x+9=-3+9 X^2+6x+9=6$ $Y:$ ...
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It's about Parabolas, I just can't seem to solve it…

Write the equation in standard form. Identify the vertex, axis of symmetry, and direction of opening of the parabola. $$x+y^2-8y=-20$$ I have seen some students answer it, but I just don't ...
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1answer
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Show that the foci of $(-8t, 4t^2)$ for varying values of $t$ is a parabola.

Show that the foci of $(-8t, 4t^2)$ for varying values of $t$ is a parabola. Also, how to attempt loci questions in general?
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1answer
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What is the approximation equation for making the day/night wave

Basically, I have a program that will graph the day/night shade similar to this page: http://www.timeanddate.com/worldclock/sunearth.html Could any of you give me the equation for graphing a line ...
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Locus of centre of circle in Lambert theorem

A beautiful theorem, when three tangents to a parabola form a triangle,the focus of the parabola lies on the circumcircle of the triangle. But what is the locus of the centre of the circumcircle of ...
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6answers
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What is one way to prove that there exists no ellipse that matches the exact curvature of the sin wave?

Preferably by not graphing both and showing they don't match visually. By the sin wave, I mean just plain old y=sinx.
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Solve system of two conic equation

I need to solve three types of system of equations in general form: System of two linear equation ($Ax + By + C = 0$) which can be done perfectly by calculating D, Dx, Dy. System of two equations ...
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1answer
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Equation of hyperbola

What is the equation of hyperbola if all axes (transverse axis, conjugate axis, principal axis) are along the coordinate axis (x and y axis), and passing through the point $(-3,4)$ and $(5,6)$. I ...
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6answers
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Finding the largest coordinate of $y$ given the ellipse $x^2+y^2-xy=1.$

The equation of the ellipse is $x^2+y^2-xy=1$. I am asked to find the the coordinate of the point $p$ on the ellipse with largest $y$ value. Thanks in advance! So I have that ...
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Equation of circle touching each axis at distance 5 units from the origin.

I have few confusions while solving the above question. First.. If circle is touching 5 units from origin then how we can find the center. In solved example it is (5,5) for the 1st quadrant. As I am ...
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1answer
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Find the equations of the circles that have centre $(0,0)$ and touch the circle $x^2 + y^2 - 8x - 6y + 24 = 0$

Find the equations of the circles that have centre $(0,0)$ and touch the circle $x^2 + y^2 - 8x - 6y + 24 = 0$ So far I have said: As the circles have centre $(0,0)$ their equations are of the form ...
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1answer
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Standard form of hyperbola

I've tried to solve this question, but I'm stuck. The question is: Find the equation in standard form for the hyperbola centered at $(5,5)$ with one focus at $(-7,5)$ and eccentricity at ...
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2answers
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Eliminate xy term of conic

For the following problem, I am trying to eliminate xy, and I've tried numerous times to solve if with no luck. I need to find the general form of the equation rotated to eliminate the xy term. $ ...
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1answer
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Finding the volume of a cone by integration of parabolic conic sections

I am working on a purely academic way of finding the volume of a right circular cone of height $h$ and radius $r$, (assume $h > r$), using integration of parabolic conic sections (conic sections ...