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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3
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13 views

Axis angle and length of ellipse

For an ellipse defined by $$x = a \cos(t + \alpha)$$ $$y = b \cos(t + \beta)$$ What are the angles and lengths of each axis? I've tried to work backwards from the expression for a rotated ellipse ...
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1answer
22 views

Physical application of conics using a ladder

Hi so I've been given a question for a Maths assignment in relation to conics and its applications. The question is: A $6m$ ladder lies against a wall. Its bottom is pulled along the floor away from ...
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3answers
32 views

how to parameterize the ellipse $x^2 + xy + 3y^2 = 1$ with $\sin \theta$ and $\cos \theta$

I am trying draw the ellipse $x^2 + xy + 3y^2 = 1$ so I can draw it. Starting from the matrix: $$ \left[ \begin{array}{cc} 1 & \frac{1}{2} \\ \frac{1}{2} & 3 \end{array}\right]$$ I ...
10
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0answers
57 views

An interesting property between a hyperbola & parabola

It is well known that when two tangents to a parabola are perpendicular to each other, they intersect on the directrix. In other words, the intersection point of the two tangents make a straight line, ...
1
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1answer
26 views

Elliptical section of a right circular cone [on hold]

A right circular cone, having cone angle $\alpha=40^o$, is thoroughly cut with a smooth plane (normal to the plane of paper as shown by the produced line AB in the diagram below) making at an acute ...
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1answer
16 views

Test if a vector is pointing towards the center of an ellipse

I have an ellipse : $$x = h + a\cos t \cos\theta - b\sin t \sin\theta \\ y = k + b\sin t \cos\theta - a\cos t \sin\theta$$ Let's say if we have a normal vector $n$ to the ellipse, on a point $p$ ...
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0answers
30 views

Two girls are 5m apart and are turning a skipping rope. [closed]

Two girls are 5m apart and are turning a skipping rope. The rope keeps a parabolic shape as it is turned. The girls keep the ends 1.2m off the ground at the ends, and the rope just touches the ground ...
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1answer
23 views

Need help with parametric parabola

so i was given my Math C assignment today and the moment i looked at question 1 i knew i had no idea what to do. This is the graph i was given (http://imgur.com/nRXOlJy). I was asked to provide an ...
3
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2answers
78 views

How to find center of a conic section from the equation?

If we are given a curve in the form $$ax^2+2bxy+cy^2+2dx+2ey+f=0$$ and the following determinant $$\delta=\begin{vmatrix}a&b\\b&c\end{vmatrix}=ac-b^2$$ is non-zero, then this is either a curve ...
2
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1answer
19 views

Conics - How to Prove

Not really sure how to approach part (iii) I have proved parts (i) and (ii), I'm assuming I have to use those answers. Any help would be greatly appreciated
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2answers
24 views

Tangents to ellipse from point outside curve

I was revising for one of my end of year maths exams, then I came across this example on how to find lines of tangents to ellipses outside the curve. Personally, I'd use differentiation and slopes to ...
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Conics Problem Help! [closed]

A triangle ABC is inscribed in a hyperbola $(2x+y-3)(x-2y+1) = 10$ . D,E,F are the middle points of sides BC, CA, AB. K & L are the circumcircles of triangles ABC, & DEF with centre at P (α, ...
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1answer
27 views

Finding angle of rotation of an ellipse

Suppose I have the ellipse $$ x^2 -2xy +4y^2 = 1 $$ How can I find the angle at which this ellipse is rotated? I have tried to assign $x=\cos\theta, y=0.5\sin\theta$ but I don't know if that's the ...
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1answer
21 views

Conic reduction

I'm trying to reduce this conic : $x^2+y^2+2xy+x+y=0$ to a canonical form. I started with finding the eigenvalues of the matrix associated to the quadratic form $x^2+y^2+2xy$ I found $z_1=2 , ...
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2answers
27 views

Finding the points of a line with a known direction and distance joining 2 ellipses

I have 2 ellipses, say $e_1$ and $e_2$. I want to draw a line $l$ connecting $e_1$ and $e_2$ in a known direction $(u,v)$, with a known distance $d$. Is there a way to solve for the points of ...
1
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1answer
24 views

Proving equations for conic sections?

How can we prove that the equations for conic sections are, indeed, sections of a cone? My guess is that it involves some sort of equality with the quadric surface equation for a cone, but I can't ...
0
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2answers
39 views

Find the slope of line L [closed]

A straight line ($L$) passing through the point $A(1,2)$ meets the line $x+y=4$ at the point $B$. If $AB=\sqrt 2$, what is the slope of $L$? With some help I did it and the slope comes out to be 2+√3 ...
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1answer
21 views

Whether or not a plane is a tangent plane to an ellipse or not? And if so, what is the point of intersection

Say we have an ellipse Transpose( p-c )A(p-c) = 1 and a plane x = a where Transpose (p-c) implies the transpose of the array p-c and a is a const A is an nxn matrix where the ellipse has ...
1
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1answer
23 views

Parabolas and lines…

Sooo... I have just received this question. 'Draw the graph of $y = x^2 + 3x - 2$'. Now, I can do this just fine. Then it says 'draw a line on the graph to solve the following equations. $x^2 + 3x ...
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0answers
32 views

Evaluate the eccentricity of the elliptical section of a right circular cone

A right circular cone, with the apex angle $\alpha=60^{o}$, is thoroughly cut with a smooth plane inclined at an acute angle $\theta=70^{o}$ with its geometrical axis to generate an elliptical section ...
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0answers
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$\text{SO}(2,1)$ is isomorphic to the group of automorphism of $x^2+y^2-z^2$

I am not able to prove the following. Let $\mathbb{K}$ be a field with more than four elements and with characteristic different from $2$. Let $\mathcal{C} \subset \mathbb{P}_2(\mathbb{K})$ be the ...
3
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0answers
29 views

Curios relation between parabola, circumcircle and circumellipse

When playing around with conics in GeoGebra, I have found out that the following relation seems to hold: Let parabola $p$ be tangent to sides/extensions of sides $BC,CA,AB$ of triangle $ABC$ at ...
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0answers
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Finding equation of line at a given angle from point to ellipse

Given a point $p_0$ and the parametric equation of an ellipse. I want to find the vector $v$ from $p_0$ such that when it intersects with the ellipse, it forms an angle $\theta$ with the ellipse's ...
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1answer
13 views

Conic Equations

I'm confused as to how you identify which equation for a conic is being used. For example, an ellipse has two equations, $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2}=1$ or $\frac{(y-k)^2}{a^2} + ...
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1answer
24 views

Projective transformation a parabola to a circle

Take the parabola $x^2 - y = 0$ in the cartesian plane. I'm not entirely sure about this, but we can express this using homogenous coordinates as $X^2 - Y = 0$ (the $W$ coefficient is $0$?) With the ...
3
votes
1answer
18 views

Unit velocity parametrization of a parabola.

I have a parametrization for the parabola $y = x^2$ given by: $$x(t) = t$$ $$y(t) = t^2$$ However, this doesn't have constant unit velocity, since $$\sqrt{x'(t)^2 + y'(t)^2} = \sqrt{1 + 4t^2} \neq ...
3
votes
4answers
81 views

How does $x^2+4xy-6x+4y^2-12y+9=0$ represent a straight line.

I need to show $x^2+4xy-6x+4y^2-12y+9=0$ is a straight line. But I only know of a straight line in the form $y=mx+c$. Any help?
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Conicoid 3-D geometry

Please make a figure explaining question :The normal at any point P of the ellipsoid x2/a2 + y2/b2 + z2/c2=1 meets the principal planes in G1,G2,G3.Find PG1:PG2:PG3.
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10 dimensional ellipsoid covering a point

In a problem I have a 10 dimensional feature space.In that feature space I draw ellipsoids with the equation transpose(x-u)*A*(x-u)=1. u is a 10 dimensional ...
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1answer
64 views

Location of the foci of a hyperbola as the value of $a$ becomes increasingly smaller than the value of $b$

"What happens to the location of the foci of a hyperbola as the value of $a$ becomes increasingly smaller than the value of $b$?" I assumed that the hyperbola was in the form ...
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2answers
39 views

Common tangent to a circle and ellipse

Hey guys i am noy able to solve this problem.So please do help me in solving this.The equation of common tangent to ellipse \begin{equation*} x^2 +2y^2=1 \end{equation*} and circle ...
0
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1answer
44 views

Why are degenerate conics not projectively equivalent to nondegenerate conics?

This is what I understand about conics being projectively equivalent. Two conics $C1=V(F)$ and $C2=V(G)$ are projectively equivalent if there is an invertible matrix $A$ such that $F(X,Y,Z)=0$ iff ...
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1answer
32 views

find equally spaced points on parabola

I'm trying to find equally spaced points on a parabola simply defined by $$y = \frac{x^2}{2 p}$$ Someone told me there is an easy way to split the parabola but he didn't tell me how and I cannot find ...
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0answers
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Roll one ellipse on another: Locus of center ever a circle?

Let $E_1$ be an ellipse fixed in the plane. Let $E_2$ be a second, possibly different ellipse, which rolls around without slippage outside $E_1$, touching perimeter-to-perimeter. Let $c_2(t)$ be the ...
2
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1answer
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Constructing a conic from two point-polar pairs

Suppose I have two points $A$ and $B$ and two lines $a$ and $b$ in the (projective) plane. Can I construct a conic section for which $a$ is the polar of $A$ and $b$ is the polar of $B$? How unique ...
3
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1answer
40 views

Find a projective transformation that sends the locus $x^2-y^2=z^2 $ to $yz=x^2$

So I'm trying to find a function that sends $x^2-y^2=z^2$ to $yz=x^2$. I know it can be done because all conics are projectively equivalent. I think I have to use a matrix of some kind but I don't ...
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0answers
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Given five points and a line find the points of the line that lie in the conic through the five points [closed]

So I'm given 5 points in general position and a line, I already know the method using Pascal's theorem to find points in the conic but I dont know how to find specifically the ones that lie on a given ...
2
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1answer
41 views

Calculating semi-minor axis of an ellipse

I'm coding a solar system animation and so far it's done, but the the orbits of the planets are circular. To make the simulation more realistic, I want to use elliptic orbits. So I visited Mercury ...
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1answer
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The use of the distance function for finding the end points of an ellipse

This is a reference to this question: Converting a rotated ellipse in parametric form to cartesian form. In the answer, it is posted that to find the extreme points of an ellipse, the distance ...
2
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3answers
28 views

Formula of parabola from two points and the $y$ coordinate of the vertex

The parabola has a vertical axis of symmetry. Given two points and the $y$ coordinate of the vertex, how to determine its formula? For example:
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2answers
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Equation of normals at the end of variable chord of parabola $y^2-4y-2x=0$

Here is my problem: If the normals at the ends of a variable chord PQ of the parabola $y^2-4y-2x=0$ are perpendicular then the tangents at P and Q will intersect at?? The correct answer is $2x+5=0$. ...
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2answers
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Transform of the Cartesian plane that maps hyperbolic arcs $xy = C$ to line segments

I have the finite set of curves: $$y = \frac{C}{x}, \qquad C = 2, 3, \ldots, C_{\max},$$ with $C$ and $x$ positive integers, $2 \le x \le C$ ($x$ varies on a finite domain). Is it possible to apply ...
2
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0answers
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Singular Conics and Intersection of Line with a Conic

I've been working through Silverman and Tate's book Rational Points on Elliptic Curves. They use conic equations as an introduction to singular/nonsingular curves. I've reproduced the problem with my ...
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0answers
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What's an elegant expression for a general conic using complex numbers?

A 'nice' form of writing a line through $2$ points $z_1,z_2\in\mathbb{C}$ is $$z\overline{z_1-z_2}+z_1\overline{z_2-z}+z_2\overline{z-z_1}=0$$ Now I'm asked to give a similar expression for a general ...
5
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1answer
36 views

Why are hyperbolas defined by two branches?

Why are hyperbolas defined by two branches, unlike a parabola which only have one? Geometrically, it looks like a slice. When plotted on a graph, it's two separate curves. Why? We were never taught ...
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0answers
21 views

Finding point on ellipse given an arc length

Given a parametric representation of an ellipse: $$ x = a\cos t \\ y = b\sin t $$ Say I have a known point $P_0$ at $t = t_0$. Given also a known arc length $d$ on the ellipse: $$ d = ...
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votes
4answers
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Find the sum of the roots of a quadratic function given the vertex of its graph

Question: At this parabola $$y = ax^2 + bx - c$$ and vertex is $T(3,9)$. What is the sum of roots of this parabola ? Help or give a hint. Thanks
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0answers
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Getting the celestial cone back from its conic section

Find the semi-vertical angle $\alpha $ of a right circular cone with z-axis symmetry cut by a plane making inclination $ \beta $ to z-axis producing the following projection on xy plane : $$ (1- ...
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vote
2answers
42 views

Affine transformation that sends a conic to itself but does not preserve the focci or the axes [closed]

So I'm trying to find an affine transformation that sends a conic to itself but does not preserve the foci or the axes. I don't know if this can be done. I'm pretty sure that if it is possible then I ...
2
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1answer
52 views

Equivalence of focus-focus and focus-directix definitions of ellipse without leaving the plane

Take a look at the following two definitions of ellipse: For some fixed points $F_1,F_2$ and real number $2a>|F_1F_2|$ an ellipse is the locus of points $P$ such that $|F_1P|+|F_2P|=2a$. ...