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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2
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3answers
71 views

Find the shortest distance between the point and a parabola

Find the shortest distance between the point $(p,0)$, where $p> 0$, and the parabola $y^2=4ax$, where $a>0$, in the different cases that arise according to the value of $p/a$. [You may wish ...
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2answers
32 views

Emulating a parabola in my game for a jump

I am currently having some trouble understanding how to plot a parabola with the x and y coordinates.In my game a player needs to jump from point a to point b and the jump would look something like ...
2
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3answers
33 views

quadratic equation plot investigation

Let $f(x) =-x^2-4x+18 $ so i plot it like this : But my imagination created the following: $-x^2-4x+18=0 -> x^2+4x = 18-> x^2+4x-18 = 0$ Which yields the parabola upside down. Where's the ...
1
vote
1answer
30 views

How do I convert these conics to standard form?

There are two conics I need to convert from general form to standard form but I am not sure if I am going about it right. They are $9x^2 + 5y^2 + 18x - 36 = 0$ and $2x^2 - 8x + y + 6 = 0$ The ...
0
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1answer
25 views

finding the vector valued function for the intersection of two functions

I asked this question yesterday and some one was nice enough to try to help me with it but after further review of the answer the function that was arived at did not seem to work when checking points. ...
3
votes
3answers
62 views

Determining angle for rotation of conics

I am working on rotation of conic sections and I'm having trouble determining the angle of rotation from the coefficients of the general conic equation. I'm given $$11x^2-24xy+4y^2+20=0$$ From this ...
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0answers
19 views

Minimize distance from a point to a hyperellipsoid

The point is outside the ellpsoid. I've found many way to solve this problem. DistancePointEllipseEllipsoid Distance-to-ellipse But I only found the numerical solution(by algorithm). Is there have a ...
2
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0answers
22 views

Hyperbola crossing, # of solutions

We typically see hyperbolas drawn the "nice" way. Namely, they are oriented with the arms "opening up" straight up or down, or at 45 degrees. But, in general, they can be at any "angle". ...
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0answers
24 views

The precise relationship between conic sections and parabolas, circles, etc. explained intuitively?

When I was first introduced parabolas/hyperbolas, circles, and ellipses, I was shown how each and every one of them could be represented as conic sections - an intersection of a plane and a conic ...
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0answers
21 views

When is the existence of rational points on an ellipse equivalent to the existence of integral points?

This question is a follow-up to my previous question. For what square-free values of $d$ is the following statement true? For all $n\geq 1$, the equation $x^2+dy^2=n$ has a rational solution if ...
2
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1answer
39 views

Existence of rational points on ellipses equivalent to existence of integral points?

Let $d$ and $n$ be square-free natural numbers. Is it true that $x^2+dy^2=n$ has a rational solution if and only if it has an integral solution? I know this is true for circles (i.e., when $d=1$) but ...
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0answers
33 views

Find bounding rectangle of arc/chord/pie cut from an ellipse

For an application, I need to know the bounding rectangle (smallest rectangle in which the object fits) for 2 objects: an arc or chord, and a pie. The objects are defined as: center (0, 0) the ...
0
votes
1answer
64 views

How to tell if (X,Y) coordinate is within a Circle

Lets say we have a circle on a MxN grid as shown below. How can we determine whether the coordinate X,Y falls within the circles coordinates under the assumptions? We know the diameter of the ...
1
vote
1answer
47 views

Finding the smallest square inside a parabola. [duplicate]

I just thought of a problem earlier today, but wanted to know if there was an easier way of acquiring the answer. Say I have a standard parabola $y=x^2$ with 3 points on it $P,Q,R$ and another point ...
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2answers
23 views

Angle between two quarters of ellipses

I must find the angle between two quarters of ellipses at their common point by the parametric equations: $R_1(t) = 3\cos (t)i + \sin (t)j$ for $0 \leq t \leq \pi/2$ and $R2(s) = \cos (s)i + 3\sin ...
2
votes
1answer
44 views

Two ways to define an ellipse

I have some problems in understanding this problem, because I'm stuck in some purely mathematical definitions and do not know how to proceed, appreciate to some that I can say which is the best way ...
2
votes
1answer
20 views

How do I find integer solutions for the following inequality

I have to write an algorithm to find all integer solutions to inequalities of the form $ax^2+by\leq c$, for example $-x^2+3y<7$, with constraints on x and y like $15\leq x\leq 31$ and $63\leq ...
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0answers
36 views

multiple parabolas repeated hotizontally

I've been trying to write/find an equation which gives me ability to introduce dips on a parabolic graph on demand, basically, change a constant value or add a piece of equation to the original one to ...
0
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3answers
88 views

How to sketch $-3x^2 - 8xy + 3y^2 = 1$ [closed]

The equation is as follows: $$-3x^2 - 8xy + 3y^2 = 1$$ How to specify the axis of the given curve? How to as accurately as possible draw a curve defined by this equation?
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2answers
33 views

What conic is $x^2-y^2-2y-2=0$ and what is the conic format of it? It would be great if work was shown so I could learn it.

What conic is $x^2 - y^2 - 2y - 2=0$ and what is the conic format of it? It would be great if work was shown so I could learn it. I know it's either a hyperbola or an eclipse but I don't know how to ...
0
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0answers
27 views

Fit an ellipse with known semi-major-axis and points

In my particular case I am given a projection of a circle onto the $xy$-plane and the radius $r$ of said circle. This results in an ellipse with semi-major axis $a$ equal to $r$. Like in this other ...
0
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2answers
33 views

How to solve this system of conics?

I am currently trying to figure out how to solve the following systems of conics: $\frac{(x+1)^2}{16} + \frac{(y-1)^2}{81} = 1$ $x+6=\frac{1}{4}(y-1)^2$ How would I find the four points that these ...
0
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1answer
40 views

Intersection of two parabolas where one is vertex shifted

I would like to be able to calculate the intersections of two parabola's which accounts for one or both of the parabola's being shifted along the x axis I have written an excel vba function to do ...
2
votes
4answers
74 views

How to find the common tangent to the curves $y^2=8x$ and $xy=-1$?

How to find the common tangent to the curves $y^2=8x$ and $xy=-1$ ? My approach: I used the formulae for tangents of a parabola and hyperbola.For any conic section if $y^2$ is replaced by ...
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1answer
39 views

vertices of a hyperbola the silliest question ever

I'm given that the center of the hyperbola is $(2,1)$ and $a=3$ and asked to find the vertices. Since vertices are on the same line with the axis of symmetry I thought the coordinates should be $(2,1 ...
3
votes
1answer
29 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
31 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
35 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 ...
13
votes
1answer
115 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, ...
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2answers
28 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 $p$ ...
0
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1answer
25 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
votes
2answers
101 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
30 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 ...
1
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1answer
32 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 ...
1
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1answer
23 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
29 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
30 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
28 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 ...
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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
54 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
14 views

$\text{SO}(2,1)$ 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
35 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
32 views

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
15 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} + ...
2
votes
1answer
31 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
23 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
88 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?
0
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0answers
18 views

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 ...