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
608 views

Calculate ellipse diameters with five points (center point and four other).

Is it possible to calculate radii (or diameters) of ellipse given: ...
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0answers
66 views

Modify ellipse equation

How can one modify an ellipse equation with a Gaussian function to get a new ellipse with bump and/or valley (example)?
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2answers
114 views

Finding An Equation For A Parabola

The information given in this particular problem: Axis is parallel to y-axis; graph passes through and $(4,11)$.$(3, 4)$ $(0,3)$ From this information, I know that it opens either upwards or ...
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2answers
239 views

Derivation Of A General Equation Of An Ellipse

I am currently reading the topic alluded to in the title of this thread. In my textbook, after the equation has been derived, $\Large\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, it says by finding the ...
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1answer
130 views

Problem with ellipse equation

How one get from this ellipse equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ that ellipse equation $$\frac{x^2+y^2}{F(\phi)^2}=1,$$ where $$F(\phi)=\frac{ab}{\sqrt{(b\cos\phi)^2+(a\sin\phi)^2}}$$ and ...
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1answer
214 views

Is the set of points of equal distance to the surface of an ellipsoid again an ellipsoid?

Consider the hyperellipsoid $A$ in $\mathbb{R}^d$ given by the semi-major axes $a_1,a_2,\ldots,a_d$. Do points on the surface of the hyperellipsoid $A'$ with semi-major axes $a_1-\varepsilon, ...
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2answers
138 views

Can an ellipse with fixed semi-axis have different values of eccentricity?

Warning: this is probably a ridiculous question but here goes... Can an ellipse with a semi-major axis $a$ take on different values of eccentricity $e$? I have seen various places where it seems to ...
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2answers
5k views

Parametric equation of a cone

I usually use the following parametric equation to find the surface area of a regular cone $z=\sqrt{x^2+y^2}$: $$x=r\cos\theta$$ $$y=r\sin\theta$$ $$z=r$$ And make $0\leq r \leq 2\pi$, $0 \leq \theta ...
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4answers
324 views

Equation of a parabola

I have trouble grasping parabolas, and mainly the cartesian equations describing the,. In my mind, there are 4 possible parabolas, a parabola shaped like a mountain ($\cap$), a parabola shaped like a ...
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2answers
2k views

Foci of Ellipse - given: Width and Height

Can you help me out with the next problem. I have an ellipse based on a width and a height. Is there any way you can find out where the focal points are? I need this information because I need to ...
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3answers
6k views

Length of a Parabolic Curve

I just wanted to know how I can find the length of a curve given by $f(x) = x^2$ from $x=0$ to $x=1$. For appproximation, the length is a bit larger than the hypotenuse of isosceles right triangle ...
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2answers
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Possibly flawed calculus homework question (tangent to ellipse)

I have an online homework program called Web Assign for my calculus course. It has given me this problem: Find equations of both the tangent lines to the ellipse x^2 + 9y^2 = 81 that pass through the ...
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2answers
58 views

How to get the center and the axes of an ellipse

Get the center and the semimajor/semiminor axes of the following ellipses: $$x^2-6x+4y^2=16$$ $$2x^2 - 4x+3y^2+6y=7$$ How would one get these? I have no clue. I have a problem with merely rewriting ...
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3answers
311 views

The equation of an ellipse

I have a couple of questions regarding ellipses. Get the equation of the ellips With Foci $(\pm 3,0)$ and which goes through $(2,\sqrt{2})$. This one I didn't understand AT ALL. I need some ...
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1answer
200 views

Equation of a parabola: Translations and directrixes

Find the equation of the paraboles, with: Focus $(3,0)$ and $x=-3$ is the directrix Focus $(0,2)$ and $y=-2$ is the directrix Vertex (I believe it is the vertex, the lowest/highest point) $(1,2)$ ...
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1answer
235 views

Apollonius ellipse equation $y^2=x \left( p-\frac{p}{2a}x \right) $ to standard form.

I am looking for a way to understand the last steps found at this site: http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=196&bodyId=203 The page finishes with showing in ...
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0answers
195 views

Parameters for an ellipse given measures of ellipticity

I am trying to visualize some data in the form: ...
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0answers
247 views

(Calculus 3) Having trouble finding the polar equation of a hyperbola.

Eccentricity e=sqrt(2), and one vertex is located at (2,0). I do know that if the vertex is located at (2,0), then the directrix is 2 units from the vertex. I am not sure how to find the location of ...
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2answers
226 views

rotation of conic sections

In the discriminant test of conic sections(rotations), why we're checking with $B^2-4AC$. How $B^2-4AC=B'^2-4A'C'$, where $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ is changed to $A'x^2+C'y^2+D'x+E'y+F'=0$ using ...
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1answer
98 views

Finding the vertices of an Ellipse

Trying to find the vertices of a ellipse. This is what I got And so used WolframAlpha just to test it out, this is my third time using it. This is the solution that I got So as you can see in ...
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1answer
81 views

Linear impluse needed for follow desired trajectory

I am trying to throw an object in my simulation with several criteria. Object is thrown from [x0,y0] object have to pass through point ...
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2answers
5k views

general equation of a tangent line to a hyperbola

Suppose that there is a hyperbola of the form $\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1$. I would like to figure out an equation that describes tangent line to this hyperbola. How would I be able to do ...
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1answer
120 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
64 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
508 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
549 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
272 views

Asymptotes of a hyperbola

Is this the correct solution something doesnot feel right
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1answer
57 views

Eccentricity of a conic

I got this solution, is this right?
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1answer
62 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
246 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
188 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
121 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
589 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
253 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
654 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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91 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
77 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
512 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
858 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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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
279 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
240 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
224 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 ...