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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1answer
116 views

Parametric Equation of a Particle Movement inside a Vortex in a Rectangular Box

I am trying to simulate the movement of a particle in a vortex in a rectangular box, I am currently using an ellipse but that causes the particle to collide with the walls more that I want. The ...
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1answer
63 views

Non-trigonometric/vector way of solving a property of a tangent line on an ellipse

Suppose that there is line $l$ that is tangent to an ellipse $A$ at point $\,P\,$. The ellipse has the foci $F'$ and $F$. One then creates two lines - each from each focus to the ...
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3answers
1k views

Proving a property of an ellipse and a tangent line of the ellipse

Suppose that there is line $l$ that is tangent to an ellipse $A$ at point $\,P\,$. The ellipse has the foci $F'$ and $F$. One then creates two lines - each from each focus to the tangency point ...
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1answer
497 views

Ellipse center with three points and the semi-axis lengths given

Having three given points in the two-dimensional plane and semi-axis lengths $a$ and $b$ of an ellipse, how to determine the center? By construction (the "Euclidean way") or analytic geometry.
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3answers
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Finding an equation of two perpendicular tangent lines of a parabola

A parabola would be given as the following: $y^2=4px$. 1) The question is, one wishes to find each equation for two orthogonal (perpendicular) tangent lines of a parabola. What would be the ...
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2answers
513 views

How to find the minimal axis-parallel ellipse enclosing a set of points.

I have a set $X$ of points in $\mathbb{R}^2$ and I'm trying to find the smallest encompassing ellipse which main axes are parallel to the coordinate system's (to put it differently, its both centres ...
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1answer
118 views

(Physics) Finding the angle(s) of launch which hit a target.

Given a coordinate and the launch speed, I need to determine which pair of angle, or angle allows a hit on said coordinate. I know, let's say, the common way, which is using the following equations: ...
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1answer
268 views

Asymptotes of a hyperbola

Is this the correct solution something doesnot feel right
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57 views

Eccentricity of a conic

I got this solution, is this right?
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1answer
61 views

Calculate the angle of a rotated conic?

I am required to calculate the rotation angle needed to come into standard form without x y product term (to make axes parallel to conic axes) in trying to find solution of problem: A conic $M$, ...
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2answers
2k views

Finding standard equation of parabola with only one vertext coordinate?

I can't seem to figure this problem out, there doesn't seem to be enough information. Find the standard equation of the parabola that has a vertical axis that has $x$-intercepts $-5$ and $3$ and has ...
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1answer
237 views

An ellipse in the rhombus

Suppose that there is an ellipse that meets with the square, but exactly inside the rhombus. The rhombus's side would be some $x$ cm. (for e.g., we can take it as $2 \ cm$.) The ellipse would have a ...
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1answer
182 views

Computing the trajectory of an orbiting body so that it collides with another orbiting body

I am creating a 2D game in which two space ships, orbiting around a planet under the influence of gravity, fire projectiles at each other, which are also under the influence of gravity. I'm creating ...
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1answer
2k views

Passing an ellipse through 3 points (where 2 two points lie on the ellipse axes)? [Updated with alternative statement of problem and new picture]

Update Alternative Statement of Problem, with New Picture Given three points $P_1$, $P_2$, and $P_3$ in the Cartesian plane, I would like to find the ellipse which passes through all three points, ...
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2answers
120 views

Why is the area of the parabolic segment $A=b^3/3$ and not $A=b^2/3$?

In Apostol's book: Archimedes made the surprising discovery that the area of the parabolic segment is exactly one-third that of the rectangle; that is to say, $A=b^3/3$, where $A$ denotes the ...
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2answers
567 views

Plotting quadratic equation in two variables

I need to draw a conic curve when quadratic equation in two variables is given: $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$ Since I can't simply check whether the pixel is solving the equation, what I'm ...
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2answers
249 views

major and minor axis of ellipse, $\phi$ (degree from $x$ axis)

The ellipse is: $$ x(t)=a \cos(wt-c)\\ y(t)=b \cos(wt-d) $$ What are: major axis length minor axis length angle of major axis with $x$ axis? the parametric form ? $(ax^2+by^2+cxy+dx+fy+...=g)$
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1answer
644 views

find the center of an ellipse given tangent point and angle

I have an ellipse with known major radius $r_x$ and minor radius $r_y$, aligned with the x- and y-axis. Given a tangent point $T$ and the tangent angle $\alpha$, how do I calculate the center $C$ ...
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89 views

What property does this equation calculate?

It's pretty difficult to Google for the meaning of a formula. This is the equation, it has to do with ellipses and GIS coordinates. $$\nu =\frac{ a} {\sqrt{(1 - (e^2 \cdot \sin(\varphi))^2)}}$$ $a$ ...
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1answer
76 views

Overlapping ellipses centered at origin:

Imagine there are two unrotated ellipses in 2d with different major and minor axes (that is to say different ellipses, but also consider case where ellipses have proportional major and minor axes, so ...
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2answers
508 views

2 circles and one ellipse and minimum area problem.

2 circles ($r_1 \neq r_2$) and one ellipse touch each other as shown in Figure-1. What is the minimum area (A) among them ? Please consider $a,b,r_1,r_2$ given values(constants). Let's imagine we ...
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1answer
840 views

Canonical form of conic section

I have $x^2+2xy-2y^2+x-4y=0$ and I have to find its canonical form, but I'm a little confused.. I'd like to understand very well what I have to do.. Can you help me, please? Thanks!
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0answers
143 views

History of calculus-based optimization

I would like to know: - who started with calculus-based optimization problems and when it was, - if there is a book focusing on history of ellipses/ conic sections - if someone ever tried to ...
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2answers
268 views

Ellipse equation parameters

If I have an ellipse expressed by: $ax^2 + 2cxy + by^2 = constant$ what does this expression equal to: $1/ \sqrt{ab - c^2}$ with respect to the ellipse ? Thanks matlabit
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2answers
236 views

The integral relation between Perimeter of ellipse and Quarter of Perimeter

Ellipse Equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ $x=a\cos t$ ,$y=b\sin t$ $$L(\alpha)=\int_0^{\alpha}\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt$$ ...
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0answers
229 views

Ellipse and circles

Playing with an ellipse I discovered the following properties and am now looking for a nice proof or references. Let's consider an ellipse with foci $A$ and $B$ and let $C$ be a point on it. Let $l$ ...
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1answer
1k views

How to calculate minimum distance between two arbitrary ellipses in 2D?

Arbitrary ellipses means that they can be scaled, translated and rotated in any way in 2D. Do you know some high-school method (might be slightly more advanced than that) to find the minimum distance? ...
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1answer
216 views

Finding vertex of a $f(x,y) = 0$ parabola

A parabola whose axis is oblique to the orthogonal coordinate axes is of the form $f(x,y)= 0$, for example $$f(x,y) = 9x^2 + 24 xy + 16 y^2 + 22x + 46 y + 9=0.$$ Using algebra only it is airly ...
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1answer
382 views

How to find intersection of an ellipse and a line that passes through the foci

There are two lines, parallel to the $x$-axis, which pass through the foci and intersect the ellipse at four points. How can I find the points of intersection? vertex: $(0,0)$ foci: $(0,10)$ and ...
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3answers
185 views

Properties of ellipse x-y form

$$5x^2+8xy+5y^2=1$$ $$1\left(\frac{x-y}{\sqrt{2}}\right)^2+9\left(\frac{x+y}{\sqrt{2}}\right)^2=1$$ I know that these two forms are equal, showing that the equation is an ellipse. I do know what ...
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1answer
514 views

Properties of ellipses and hyperbola related to matrix operation

$5x^2+8xy+5y^2=\mathbf{x}^TA\mathbf{x}= > (S^T\mathbf{x})^TD(S^T\mathbf{x})=1\left(\frac{x-y}{\sqrt{2}}\right)^2+9\left(\frac{x+y}{\sqrt{2}}\right)^2$ Thus, the equation is that of an ellipse, ...
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1answer
682 views

Finding the major and minor axes of an $n$-dimensional ellipse

Here are two n-dimensional vectors: $V_1$ and $V_2$ $V_1 (v_1,v_2, \dots ,v_n)$ $V_2 (v_1,v_2, \dots ,v_n)$ $V_1 \cos(\theta) + V_2 \sin(\theta)$ is an ellipse in the $n$-D space. (Its center is ...
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3answers
764 views

Ellipse with non-orthogonal minor and major axes?

If there's an ellipse with non-orthogonal minor and major axes, what do we call it? For example, is the following curve a ellipse? $x = \cos(\theta)$ $y = \sin(\theta) + \cos(\theta) $ curve ...
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1answer
198 views

Nearest point on an ellipse in n-dimensional space.

Here are two n-dimensional vectors: $V_1$ and $V_2$ $V_1 (v_1,v_2, \dots ,v_n)$ $V_2 (v_1,v_2, \dots ,v_n)$ It seems that $V_1*cos(\theta) + V_2*sin(\theta)$ is an ellipse in the n-D space. (Its ...
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2answers
3k views

Finding a point on a ellipse so that it has the shortest distance between this point and another given point [duplicate]

Possible Duplicate: Calculating Distance of a Point from an Ellipse Border Given a point $A = (x_1, y_1)$ and a $2$D ellipse, how could we find a point $B = (x_2, y_2)$ on the ellipse so ...
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1answer
761 views

Parametrization of a conic and rational solutions

How can we parametrize the conic $C$: $x^2+y^2 = 5$, by considering a variable line through $(2,1)$ and hence all rational solutions of $x^2 + y^2 = 5$? I'm thinking let $x = \sqrt{5}\cos t$, and $y ...
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1answer
254 views

Finding the conjugate hyperbola

Assume we are given a general proper hyperbola $ a_{11}x^2 + 2 a_{12} x y + a_{22} y^2 + 2 a_{13}x +2 a_{23}y + a_{33} = 0$ with $D =\det \begin{pmatrix} a_{11} & a_{12} \\ a_{12} & ...
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1answer
169 views

Given the width and height of an ellipse find n number of points around the ellipse?

I am looking to find $n$ number of points around an ellipse. They don't necessarily have to be equidistant. Similar to what this forum is asking: I found several answers that are similar but I am ...
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3answers
69 views

Describing the effect on $ax^2$ by manipulating $a$

Please take, for example, $y = x^2$ and $y = 2x^2$. Graphs: Wolfram Alpha What is the most appropriate way to describe the effect of $a$? "$a$ causes the parabola to open at $1/a$ the rate of $y = ...
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2answers
112 views

Why is the landing footprint an ellipse?

Following the Curiosity landing I noticed that the possible landing site (the so-called 'landing footprint') was demarcated by an ellipse. Here is a picture of it: Now obviously such a footprint ...
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0answers
128 views

How to calculate the position of the (right circular) cone's vertex projection upon an ellipse?

Starting with a plane parallel to the cone's base (defining a circle with center O), the plane is rotated by φ degrees around one axis that contains the center of the original circle (that is, around ...
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1answer
353 views

Equation of the locus of centre of the ellipse?

An ellipse slides between two perpendicular lines. To which family does the locus of the centre of the ellipse belong to?
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1answer
462 views

Tracing points around a curve/ellipse

Sorry if this has been asked before but my maths days are long behind me. What I want to know is how to find out the coordinates along the circumference of an ellipse. So supposing I am at point ...
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3answers
977 views

Normal to Ellipse and Angle at Major Axis

I've tried to detail my question using the image shown in this post. . Consider an ellipse with 5 parameters $(x_C, y_C, a, b, \psi)$ where $(x_C, y_C)$ is the center of the ellipse, $a$ and $b$ are ...
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3answers
582 views

How to draw an ellipse with its center and two points on it with with Sketchpad?

How to draw an ellipse with its center and two points on it with Sketchpad?
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1answer
531 views

Calculating equidistant points around an ellipse arc

As an extension to this question on equiangular fisheye distortion, how can I calculate equidistant points around an ellipse (or 1/4 segment of) given it's aspect ratio? When it's circular, I can use ...
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1answer
786 views

A hyperbola passing through integer lattice points

Prove that for any $n\geq 0$, there is a hyperbola that passes through exactly $n$ lattice points (= points with integer coordinates) and find an example. For example it is easy to see that the ...
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1answer
518 views

How do I get a tangent to a rotated ellipse in a given point?

I have just graduated from a school you would call High School and even though we talked about tangents to ellipses, we never covered rotated ellipses. So, what I am looking for, is a formula for a ...
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1answer
1k views

How many points are necessary to find a parallel ellipse, and how to do it?

So, I understand that to find an ellipse for sure you need at least five points. Why? The ellipse equation has only four variables ($x_0, y_0, a,\text{ and }b$). That's not actually my true question, ...
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3answers
7k views

Finding Asymptotes of Hyperbolas

To find a asymptote its either b2/a2 or a2/b2 depending on the way the equation is written. With the problem $$\frac{(x+1)^2}{16} - \frac{(y-2)^2}{9} = 1$$ The solutions the sheet I have is giving ...