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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Need help with the proof of conic section

Prove that the intersection of a plane and a object consist of one cone and one upside-down cone where the tip of cone meet is either degenerate conic or conic Also, idenify in what situation, the ...
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71 views

Some question concerning curve of second order

Let $$F(x,y)=ax^2+2bxy+cy^2+2dx+2ey+f,$$ $$\phi(x,y)=ax^2+2bxy+cy^2,$$ $x,y \in \mathbb{R}$. Assume that for some $x_0, y_0 \in \mathbb{R}$ and for some $\alpha, \beta \in \mathbb{R}$ such that ...
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1answer
135 views

Parabola with a variable starting point

I am trying to build an equation where I could start at (x,y) which are known and create a parabola from that starting point. I have no idea where it intercepts the X or Y. I know where I want the ...
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1answer
393 views

Properties of Parabola / Optimization

I've been working through some past papers for an exam which I am due to be sitting tomorrow. In the Conic Sections paper from a couple of years ago, the following question came up: The path of a ...
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1answer
3k views

how to find the parabola of a flying object [closed]

how can you find the parabola of a flying object without testing it? what variables do you need? I want to calculate the maximum hight and distance using a parabola. Is this possible? Any help will be ...
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4answers
244 views

Difficult equations to rewrite as ellipses

I have this equality that defines an elliptic boundary. I am trying to rewrite it in the form of the equation of an ellipse, but I am having trouble doing that. How would I go about rewriting this ...
6
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1answer
918 views

Relationship between ellipsoid radii and eigenvalues

I should start by saying that I haven't done algebra for very long time. I recently have some work related to algebra, so I need some help to speedup. I went through a theorem in the book stating the ...
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1answer
499 views

Center of gravity of an ellipse

I think the center of gravity of an ellipse is the intersection point of it's two radius. But I didn't see it anywhere, so I'm having some doubt about it. Am I right? Thanks to all.
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4answers
2k views

Is an ellipse a circle transformed by a simple formula?

Does any ellipse $E$ have a circle $C$ such that you can obtain $E$ by transforming $C$ by a simple formula $F$? In details , both $E$ and $C$ have the same center and the axes of $E$ are the XY axes. ...
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1answer
204 views

How to compute the cross point between an ellipse and a straight line?

Please let me know how to compute the possible cross points between an ellipse and a straight line. In details , I know the basic properties of the two shapes. So if the ellipse had its center at the ...
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2answers
180 views

Hyperbola property

I am posting the following question under homework category. I hope I will have very good answer from mathematicians about conic sections. I have seen closely the conic sections and their ...
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3answers
14k views

Check if a point is within an ellipse

I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane. How do I determine if a point $(x,y)$ is within the area bounded by ...
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3answers
867 views

an important property of an ellipse

Good morning everybody. I would like to know the proof of the following observation on the ellipse. A circle is drawn with the right latus rectum as diameter. Another circle is drawn with its ...
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3answers
289 views

Apostol Section 13.25 #13 - Conic Sections

Question: Prove that a similarity transformation (replacing $x$ by $tx$ and $y$ by $ty$) carries an ellipse with center at the origin into another ellipse with the same eccentricity. (The next ...
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2answers
335 views

Maximize the distance between a line normal to an ellipse and its center

My friend sent me this problem, which (upon Googling) seems to be from a Cornell class (1220?). Anywho. My advice to him was to parametrize the ellipse (say, in the first quadrant) with $x = a ...
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1answer
752 views

A Hunt for a Mathematical Machine That Gives Points

The central question is : Is there any method for Producing the global Points on the curve (any cubic curve, or at least a Degree-2 curve ) , if we have local Part with us ? Explanation: ...
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2answers
318 views

why we only have a approximation for every circumference for ellipse but not define a special formula for each ellipse

Why do we only have an approximation for every circumference for ellipse, but we cannot define a special ratio formula for each ellipse? Is it possible for people to use a computer to find the exact ...
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1answer
298 views

How to find points of tangency on a hyperbola?

If tangent lines to the hyperbola $9x^2-y^2=36 \;$ intersect y-axis at point $(0,6)$, find the points of tangency.
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4answers
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How to find points of tangency on an ellipse?

The problem I have to solve is: If tangent lines to ellipse $9x^2+4y^2=36$ intersect the y-axis at point $(0,6)$, find the points of tangency.
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5answers
535 views

Generating coordinates for 'N' points on the circumference of an ellipse with fixed nearest-neighbor spacing

I have an ellipse with semimajor axis $A$ and semiminor axis $B$. I would like to pick $N$ points along the circumference of the ellipse such that the Euclidean distance between any two ...
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1answer
144 views

How to “transform” $(f(x), g(x))$ to $(x, y(x))$?

I'm currently trying to solve the following problem: Let $L$ be the set of points of $\mathbb{R}^2$ that satisfy the condition $f(x,y) = 7x^2-6 \sqrt{3} xy + 13y^2 = 16$. It is possible to ...
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2answers
332 views

Why do definitions of distinct conic sections produce a single equation?

I understand how to get from the definitions of a hyperbola — as the set of all points on a plane such that the absolute value of the difference between the distances to two foci at $(-c,0)$ and ...
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3answers
831 views

Parametric form of an ellipse given by $ax^2 + by^2 + cxy = d$

If $c = 0$, the parametric form is obviously $x = \sqrt{\frac{d}{a}} \cos(t), y = \sqrt{\frac{d}{b}} \sin(t)$. When $c \neq 0$ the sine and cosine should be phase shifted from each other. How do I ...
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2answers
791 views

What is the simplest ellipse that goes through exactly 13 lattice points?

The ellipse $-30 x + 3 x^2 - 10 y - 3 x y + 4 y^2$ goes through exactly 11 lattice points. Another such ellipse is $4 - 30 x + 2 x^2 - 5 y - x y + 3 y^2$. What is the simplest ellipse that goes ...
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1answer
299 views

Irreducible conic implies that the underlying matrix is invertible

I guess that it is true that a conic (2nd degree homogeneous equation in complex variables) is irreducible (i.e can't be factorized over polynomials) if and only if the underlying matrix of ...
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2answers
939 views

Plot ellipse in cartesian coordinates

I am working with a computer program that needs to draw objects in a 3d field. Over time, these objects will move (at different rates) on an elliptical path around the 0,0,0 point. I've read the ...
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1answer
153 views

Does using an ellipse as a template still produce an ellipse?

Suppose I have a (physical) template, consisting of a piece of stiff sheet plastic with a hole cut in the middle. Suppose the hole is in the shape of an ellipse, say, 8 x 12 inches. Suppose I then ...
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2answers
303 views

Finding upper segments of intersecting parabolas

I have multiple parabolas ($y = ax^2 + bx + c$) which may intersect with each other (or some of them may not intersect). I am trying to find upper segments of these parabolas, e.g. bold part in the ...
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2answers
3k views

How to draw an parabolic path between two points

I have two sets of points and i want draw an parabolic arc between two points and also to find the intermediate points which the parabolic path is drawn.... In the above image you can see the curve ...
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1answer
1k views

Formula for curve parallel to a parabola

I have a simple parabola in the form $y = a + bx^2$. I would like to find the formula for a curve which is parallel to this curve by distance $c$. By parallel I mean that there is an equal distance ...
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3answers
1k views

Canonical to Parametric, Ellipse Equation

I've done some algebra tricks in this derivation and I'm not sure if it's okay to do those things. $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = \cos^2\theta + ...
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3answers
857 views

About the discriminant for ellipse in the implicit polynomial

It is known that a conic can be expressed as an implicit second-order polynomial as follows: $$au^2 + buv + cv^2 + du + ev + f = 0$$ where $(u,v)$ is the 2d coordinate of the point on the conic. If ...
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3answers
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Are there parabolic and elliptical functions analogous to the circular and hyperbolic functions sin(h),cos(h), and tan(h)?

In matters of conic sections, are there other properties such that it helps to group the circle and hyperbola in one, and the parabola and ellipse in the other?
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4answers
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Tangent of an ellipse to an outside point

Let $C$ be a curve that is given by the equation: $$ 2x^2 + y^2 = 1 $$ and let P be a point $(1,1)$, which lies outside of the curve. We want to find all lines that are tangent to $C$ and intersect ...
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4answers
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How to partition area of an ellipse into odd number of regions?

Is it possible to divide an ellipse into 3,5 or 7 etc. parts of equal area? If yes then how? Describe a circle around the ellipse and the circle of an equilateral triangle we construct. Projection ...
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0answers
368 views

How to find the center of an (scaled) ellipse?

This question is an extension of How to find the center of an ellipse?. The solution there works well, but in Javascript the floating point calculations are not that accurate. The workaround is to ...
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0answers
388 views

Best fit hyperbola with specific constraint

Given a set of points $(x_i, y_i)$ in $\mathbb{R}^2$, I can find the best fit hyperbola in the least squares sense by using the method given here. But, is there a way to constrict the hyperbola to ...
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1answer
316 views

How do I calculate $t$ in the general parametric equation of an ellipse when the point $(x,y)$ is known?

I have the general parametric equation of an ellipse. $$\begin{align*}x&=c_x+a\cos{t}\cos{\alpha}-b\sin{t}\sin{\alpha} \\ y&=c_y+a\cos{t}\sin{\alpha}+b\sin{t}\cos{\alpha}\end{align*}$$ I ...
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1answer
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What is the equation for 3D analog of the parabola (the paraboloid) using cartesian co-ordinates?

In 2D, the equation is: $y=4a(x-x_0)^2+c$ What is the equation for 3D analog of parabola?
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700 views

How to find the center of an ellipse?

I have the following data:- I have two points ($P_1$, $P_2$) that lie somewhere on the ellipse's circumference. I know the angle ($\alpha$) that the major-axis subtends on x-axis. I have both the ...
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3answers
198 views

What is the most direct way to derive an equation for a parabola from its x and y intercepts?

I have a pair of points at my disposal. One of these points represents the parabola's maximum y-value, which always occurs at x=0. I also have a point which represents the parabola's x-intercept(s). ...
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1answer
468 views

Property of an ellipse

I need proof for the following question. Also, I want to know, can we apply the same for other conics. If yes, where and when... Please explain. Show that there exists a point K on the major axis of ...
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3answers
780 views

A hyperbola as a constant difference of distances

I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is $2a$, the distance between the two ...
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3answers
759 views

Parabolic shape in Bow (not arrow!)

This is what I am thinking for some days. And I think here are some experts who can answer this question. If I bend any stick made with material that uniform density and its shape is cylindrical ...
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3answers
626 views

Visualization of complex roots for quadratics

I read that if a parabola has no real roots, then its complex roots can be visualized by graphing the same parabola ($ax^2 + bx + c$) with $-a$ and then finding the roots of that, then using those ...
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1answer
437 views

Foci of a general conic equation

The general equation of a conic is $A x^2 + B x y + C y^2 + D x + E y + F = 0$. At Wikipedia, there is an equation for the eccentricity, based on ABCDEF. Is there a similar equation for getting ...
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1answer
148 views

Roots of parabola

I have a parabola with the equation $y = x^2 + 6x + 7$ and I am trying to calculate the $x$-intercept points. Here is my working so far... let $y = 0$, $x^2 + 6x + 7 = 0$ $x(x + 6) = -7$ After ...
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1answer
627 views

Generalization of ellipse equation to higher dimensional surfaces

This question was motivated by Definition of an ellipsoid based on its focal points . I'd like to avoid terms like ellipsoid, so I'll use terms like one-dimensional ellipse (a normal ellipse in the ...
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Can Fermats descent be interpreted on a conic?

Fermat proved the Diophantine equation $$(x^2)^2 + (y^2)^2 = z^2$$ has only solutions $(0,0,0)$, $(0,\pm 1,\pm 1)$ and $(\pm 1,0,\pm 1)$ using "infinite descent". The conic $C : X^2 + Y^2 - 1$ has a ...
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1answer
681 views

Confusion with the various forms of the equation of second degree

I am confused with the second degree equation,an equation of second degree $ax^2+by^2+2hxy+2gx+2fy+c=0$ represents a conic,and nature of the conic depends on the various other conditions,like if ...