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 scale x- and y- axes equally in Maple?

I have the ellipse $\frac{25}{36}x^2+\frac{5}{36}y^2=1$. Maple draws it as a circle: How can I change the coordinates, to make it look like an actual ellipse?
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Plane and Ellipse Intersection

Short Version: If some can solve the easier to read form as follows, I would be thankful. \begin{equation} B = \frac{1 - d^{T}Bd}{ K_{1} } A \end{equation} \begin{equation} B^{T}d = \frac{1 - ...
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1answer
31 views

Parabola Application [on hold]

A cannonball when configured to be fired at a certain angle would have a parabolic path with a maximum height of $50\mathrm{m}$ and a horizontal range of $20\mathrm{m}$. If the cannonball is placed at ...
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1answer
19 views

Conic-Sections, Ellipse, Parametrics, Normals and Tangents [on hold]

Find the equation of the tangent and normal to the ellipse defined parametrically by: $x=5\cos\theta$ and $y=3\sin\theta$ at the point where $\theta=\pi/4$.
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Projective conic generated by a set of tangent triangles.

I need to proof the following result: Let C be a real projective conic and P, Q two points interiors to C then there is another real projective conic such that every triangle inscribed on that conic ...
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1answer
36 views

Attempting to put the following conic into standard form: $y^2-2x^2+8y-8x-4=0$

Put the following conic into standard form: $y^2-2x^2+8y-8x-4=0$ I ended up with $$ -\frac {(x-2)^2}{12} + \frac{(y+8)^2}{24} = 1, $$ but I'm not sure if this is right.
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1answer
57 views

Find the differential equation of parabolas with axis parallel to the x-axis [closed]

Can someone please help me find the differential equation of this problem? I can't think of any solution to start with and kindly show your graph. I was told that the answer is $y'y'''-3(y'')^2=0$
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Strong duality in conic programmig

Let $K$ be a convex cone which is not closed. The look at a probem of the from $$\min <C,X>, \,s.t\; <A_i,X>=b_i,\, X\in C.$$ Now suppose I now that both this program and its dual have a ...
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8answers
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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 ...
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1answer
12 views

Cone Plane Intersection Radius Size

I have a cone which is passing through a plane. The cone is not perpendicular to the plane, so the intersection area between the cone and the plane will not be circular but an ellipse. The cone will ...
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Differential Equations: Confocal Ellipse and Hyperbola

I am currently brushing up on Conic Sections, and I am having some problems on solving a first order quadratic differential equation. I would appreciate any help on the topic! I know that confocal ...
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2answers
30 views

Questions on the relation of the axis of a cone to its conic sections

(1) Does the axis of a cone pass through the foci of any its conic sections? Consider the image below: Is the intersection of the axis of cone and the ellipse the same as the focus of the ellipse? ...
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3answers
42 views

Is this equation a parabola or a hyperbola?

In a 1972 paper by Robert Merton, the following equation is derived: $$\sigma(\mu;A,B,C,D)=\sqrt{\frac{A \mu^2-2B\mu+C}{D}}$$ This is known as the Markowitz frontier in finance. When this is ...
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0answers
49 views

Is the following a conic section

All vectors are in $\mathbb{R}^3$ and only $\mathbf{r} = \left[ x; y; z \right]$ is unknown. My question is does the following system define a conic section in the $x-y$ plane and, if so, how can I ...
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1answer
19 views

Location of an arbitrary point of an ellipse

Given this ellipse equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $(a>b>0)$ and $c:=\sqrt{a^2-b^2}$ aswell as the focal points $F=(c,0)$ and $F'=(-c,0)$, why can we say without loss of ...
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0answers
40 views

Solving Kepler's Equation

I've been working on simulating orbits. I've found that, when solving Kepler's equation, $M = E - \varepsilon\sin{E}$, I'm unsure about the solution to use. For a true anomaly $< \pi$, using the ...
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2answers
43 views

Classification of conics in hyperbolic plane

How many different types of conics exist in hyperbolic plane? Euclidean geometry has three, of course. But when I was trying to find out results for the hyperbolic plane, the best thing I found ...
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1answer
23 views

Parametric equations - locus at midpoint

Consider the parametric equations $x=-2t^2$ and $y=4t$ The normal at any point, P, cuts the x-axis at Q. Find the Cartesian equation of the locus of the midpoint, M, of PQ. Can anyone help get me ...
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3answers
45 views

Equation of a parabola that passes thorught 2 point with know slopes

I want to be able to solve for the equation of this parabola. Known Points A(2,1) Slope @ A=1/2 B(7.25,2.5) Slope @ B=1/5 nothing else is known/given, The picture shows that parabola's Axis of ...
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Visualising 3rd degree equations

I know that general second degree curve, i.e. $ax^2 + by^2 + 2hxy + 2gx + 2fy + c=0$ gives us the equation of different cross sections of a cone. Similarly, what does a third degree* curve actually ...
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Relationship between polar angle and tangent angle of a conic section

I'm trying to define the relationship between the polar angle (or gradient) of a conic section and the tangent angle (or gradient) without resorting to x,y coordinates of the tangent point, i.e. given ...
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2answers
79 views

What's interesting in latus rectum?

I'm a maths teacher in Italian secondary school and I've been spending some time trying to construct "meaningful" problems about conic sections. I particularly like problems which focus on practical ...
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1answer
29 views

A parabola with a horizontal directrix has it's focus at (2,5). If the point (7,-7) is on the parabola:

A. Find a possible equation for the directeix. B. Using your results from A. Find the vertex. C. Write an equation for the parabola This question is giving me some major difficulty I have spent 3 ...
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1answer
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Approximating the circumference of given ellipse

Say we got the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{24}=1$, and the goal is to find the circumference using line integrals. So I parametrized the curve by $x=5\cos(t)$, $y=\sqrt{24}\sin(t)$. By ...
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1answer
29 views

What do you get when you take a conic section in between a parabola and vertical?

The way conic sections are often described, if you take a section parallel to the double-cone, you get a parabola, and if you take a perfectly vertical section, you get a hyperbola. But what if you ...
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3answers
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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
29 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 ...
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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 ...
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1answer
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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 ...
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1answer
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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. ...
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3answers
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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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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 ...
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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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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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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 ...
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1answer
32 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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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 ...
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1answer
62 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 ...
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1answer
44 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
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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 ...
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1answer
43 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 ...
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1answer
19 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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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 ...
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3answers
82 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
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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 ...
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0answers
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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 ...
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2answers
31 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 ...
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1answer
33 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 ...
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4answers
64 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 ...