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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Interpreting 3D parametric equations

I've been working through a problem and I have managed to reduce it to the following:$$x=\frac{2r}{3}\cos\theta - \frac{r}{3}\sin\theta$$ $$y=\frac{2r}{3}\sin\theta - \frac{r}{3}\cos\theta$$ $$z = ...
2
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3answers
742 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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1answer
1k views

How can convert the general form of ellipse equation in the standard form?

How can convert the general form of ellipse equation in the standard form? $$-x+2y+x^2+xy+y^2=0$$ Thank you in advance?
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1answer
9 views

The focal chord that cuts the parabola $ x^2 = -6y$ at $(6, -6)$ cuts the parabola again at $X$

The focal chord that cuts the parabola $x^2 = -6y$ at $(6, -6)$ cuts the parabola again at $X$. Find the coordinates of $X$. I have been going insane someone please help me :(
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2answers
44 views

Calculating semi axes from given tilted ellipse equation

Hopefully no duplicate of Ellipse $3x^2-x+6xy-3y+5y^2=0$: what are the semi-major and semi-minor axes, displacement of centre, and angle of incline? (see below) Let the following equation $$x^2 - ...
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0answers
26 views

What is the size of the opening of a parabola?

What variable affects the size of a parabola in vertex form? Please help me, this is a school homework. Tthank you s much for your help.
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0answers
12 views

Dandelin spheres and the asymptotes of a hyperbola

The other day, I was reading up on the synthetic geometry of conic sections a bit, and I wondered: is it possible to construct the hyperbola's asymptotes given just the intersecting plane and the ...
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2answers
113 views

Conics generalized to surfaces of constant curvature

Do conic sections have an interesting generalization to surfaces of constant curvature? Consider a sphere (constant positive curvature) $\mathcal{S}$ centered at $O$, as well as points $A, B \in ...
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2answers
62 views

Fit an ellipse with constraints

I'd like to fit an ellipse with the equation of $ x^2 + ay^2 + bx + c =0 $ This is basically the equation of an ellipse with no tilt and with its center on the horizontal axis. I have some ...
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0answers
28 views

Area swept out by non-solar focus not same over equal time?

Per Kepler's laws, the area swept out by a line between the sun and a planet is equal for a given period of time. The sun is also one focus of the planet's elliptical orbit. What about the area swept ...
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2answers
3k 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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2answers
40 views

Ellipse with center in origin

The purpose is to fit data to a ellipse which center is the origin $(x_0=0,y_0=0)$. I found the general quadratic curve: $$ax^2+2bxy+cy^2+2dx+2fy+g=0$$ Reference: ...
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1answer
23 views

Funny interconnection between a triangle and the ellipse inscribed

Le $p\in\Bbb R[X]$ be a 3rd degree polynomial. Suppose it has one real root and two complex conjugate roots: these three points forms a triangle in the complex plane. Consider the ellipse inscribed ...
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4answers
41 views

Show that if an ellipse and a hyperbola have the same foci, then at each point of intersection their tangent lines are perpendicular.

I have to show that: If an ellipse and a hyperbola have the same foci, then at each point of intersection, their tangent lines are perpendicular. So I know that if I prove it for one of the ...
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0answers
11 views

Equivalence of definitions for a conic

I have to prove that these two definitions for the eccentricity of a conic $C$ are equivalent: Ratio between the distance of the points $x$ in $C$ to $f$ its foci and $l$ its directrix. Ratio ...
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0answers
6 views

Parametric equation of the horizontal Hyperbola

I have to show that the parametric equation of the horizontal hyperbola is given by: $$ x=a \sec\theta \\ y=b \tan \theta $$ where $a$ and $b$ are the distance between the centre and the foci and ...
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0answers
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Development of intersection of two cones and two planes.

(Crossposted on http://www.boatdesign.net/forums/boat-design/development-intersection-two-cones-two-planes-51910.html#post713541 ) It is well known that a cone is a ...
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1answer
29 views

Rolling ellipse on line - tangent and normal of roulette

Suppose that an ellipse is rolling along a line. If we follow the path of one of the foci of the ellipse as it rolls, then this path formes a curve - namely an undulary. Now consider the following ...
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0answers
62 views

Can one prove Brianchon's theorem using Ceva's theorem?

Can I prove Brianchon's theorem using Ceva's? I am also wondering if parabola and hyperbola can be inscribed in a hexagon?
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1answer
32 views

Find a specific rectangle in an ellipse

For a software developpment, I need to find a rectangle that fits in an ellipse. I have an outer rectangle (left, top, width and height) and a function that draws an ellipse in it. Now I need to know ...
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1answer
11 views

finding parabola equation by angle and 2 points [closed]

ok so I got a kind of mechanic question. I got two point (0,0) and (8,0) (8 meters between), I got an angle at X=0 of 20 degree and I got deaccelaration of 10 m/sec^2. how can I find the max point of ...
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3answers
88 views

Ellipse $3x^2-x+6xy-3y+5y^2=0$: what are the semi-major and semi-minor axes, displacement of centre, and angle of incline?

Given the ellipse $$3x^2-x+6xy-3y+5y^2=0$$ find the following: semi-major axis, $a$ semi-minor axis, $b$ displacement of centre from origin (or coordinates of centre of ellipse $(h,k)$) angle of ...
2
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1answer
179 views

Math function for parabola

I need an implicit function that plots the parabola that I am showing you in the picture. Everything you need is shown there. The radius of the thickness of the parabola must be 3. Thank you in ...
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2answers
9k views

How to determine the arc length of ellipse?

I want to determine the arc length of a ellipse. So what data should I know ? And what law should I use ? For example I have this ellipse on picture below: How can I determine the $d$ length of ...
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9answers
3k views

Why is an ellipse, hyperbola, and circle not a function?

I am aware of the vertical line test. If you place a vertical line over a shape, and if it crosses more than once, it fails the vertical line test and is no longer a function. But I don't understand ...
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1answer
16 views

Grade 10 Quadratic equation

This was on my year 10 maths test and I gave up with 40 mins to complete: Basically you were given the coordinates: y intercept : (0,10) 1 x intercept: (10,0) and y value of the vertex: +15 Can ...
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2answers
75 views

Concurrency-Three parabolas sharing common directrix.

I have found this result by exploring for new problems. If three parabolas share a common directrix and each pair intersect each other in two points, then, the lines joining the two intersection ...
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0answers
25 views

Show a curve has no factor of degree 1 or 2

I have to show that $ h(x,y)=y^{2}(x^{2}+x+1)-x^{2} $ has no factors of degree 1 or 2. I know that h contains infinitely many points and is singular at the points (1,0,0), (0,1,0) and (0,0,1). I am ...
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1answer
18 views

How to convert formulas for different standard parabolas?

There are 4 types of standard parabolas , and I'm supposed to remember many formulas about them like tangent , normal etc. But the problem is , if i know a certain formula for $y^2=4ax $ how can i ...
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1answer
19 views

help needed in understanding general conics proof

The origin is a centre of a general conic of second degree iff the coefficients of linear terms vanish. $ (\Rightarrow)$ part: Let $$ Q(x,y)\equiv ax^{2}+2h xy+ by^{2} + 2gx+2fy+c=0$$ books ...
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1answer
244 views

Determining the direct and transverse tangent lines for two non-overlapping ellipses

I am trying to determine the direct and transverse lines for two non-overlapping ellipses. I specifically mean that the two ellipses are totally separated from each other with no shared regions. I ...
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4answers
2k views

Parabola is an ellipse, but with one focal point at infinity

While I was reading about conic sections, I came across the following statement: A parabola is an ellipse, but with one focal point at infinity. But it is not clear to me. Can someone explain it ...
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1answer
33 views

Arc length of parabola between two points

Well lets take a parabola of the equation $y = f(x)$ where $f(x)$ is obviously a $2^{nd}$ degree function. Now lets take two points at $x=a$ and $x=b$ . So can anyone please help me to find that ...
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1answer
67 views

Focus of a parabola, without derivatives

I have a seemingly easy question, but I have no clue how to find out its answer. I have the function $$f(x)=\tfrac{1}{8} x^2$$ This function is for (a parabolic cross-section through) a paraboloid ...
4
votes
2answers
51 views

Better substitution calculating integral?

I'm calculating $$ \iint\limits_S \, \left(\frac{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}{1+\frac{x^2}{a^2}+\frac{y^2}{b^2}} \right)^\frac{1}{2} \, dA$$ with $$S =\left\{ (x, \, y) \in \mathbb{R}^2 : ...
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1answer
14 views

transformation of conic section

Given is the conic section $x^2 +xy + y^2 +2x +3y -3 = 0$. The following tasks: 1.) What is the coordinate matrix $A_1 = M_{\beta} (\sigma) $ of the bilinearform? 2.) do the transformation and ...
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5answers
2k views

How to calculate ellipse sector area *from a focus*

How do you calculate the area of a sector of an ellipse when the angle of the sector is drawn from one of the focii? In other words, how to find the area swept out by the true anomaly? There are ...
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1answer
22 views

Given area of sector and a starting angle from focus of an ellipse, finding angle needed to get area.

Problem Background: I'm trying to make a rough simulation of Kepler's second law (equal areas over equal time) and to do this I've divided the area of the ellipse into some number of pieces. I want ...
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2answers
413 views

Equal angles formed by the tangent lines to an ellipse and the lines through the foci.

Given an ellipse with foci $F_1, F_2$ and a point $P$. Let $T_1, T_2$ the points of tangency on the ellipse determined by the tangent lines through $P$. Show that $\widehat {T_1 P F_1} = \widehat {T_2 ...
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1answer
338 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
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1answer
358 views

How to compute the chord length of an ellipse?

How do I calculate the chord length of ellipse? I need to design a blade vane rotating inside an elliptical profile. The blade is supposed to be in contact with ellipse with a maximum clearance of ...
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1answer
667 views

length of the focal chord

Paragraph: $PQ$ is a focal chord of the parabola: $y^2=4ax.$ The tangents to the parabola at the points $P$ and $Q$ meet at point $R$ which lies on the line $y=2x + a.$ Question: Find the length ...
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1answer
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Conic involving circle question. [duplicate]

The question is: If the curves $ax^2+4xy+2y^2+x+y+5=0$ and $ax^2+6xy+5y^2+2x+3y+8=0$ intersect at four concyclic points then the value of a is???? The options are: a) 4 b) -4 c) 6 d) -6 I've ...
2
votes
1answer
18 views

Graphing Circles, Ellipses, Parabolas, and Hyperbolas

I need help plotting a curve on a graph where the distance from focus1 is always the same ratio to the distance from focus2. For instance, lets assume focus1 is -5 along the x axis, and focus2 is +5 ...
6
votes
1answer
896 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 ...
3
votes
5answers
74 views

Find the minimum distance between the curves $y^2-xy-2x^2 =0$ and $y^2=x-2$

How to find the minimum distance between the curves $y^2-xy-2x^2 =0$ and $y^2=x-2$ Let $y^2-xy-2x^2 =0...(1)$ and $y^2=x-2...(2)$ In equation (1) coefficient of $x^2 =-2; y^2=1, 2xy =\frac{-1}{2}$ ...
2
votes
2answers
37 views

What's the parametric equation for the general form of an ellipse rotated by any amount?

What's the parametric equation for the general form of an ellipse rotated by any amount? Preferably, as a computer scientist, how can this equation be derived from the three variables: coordinate of ...
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3answers
29 views

Find equation of tangents to hyperbola

$$\frac{x^2}{4} - \frac {y^2}{16} = 1$$ There is a point $(1,2)$ where $2$ lines pass through and are a tangent to both curves. How do I find the equation of both lines?
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1answer
21 views

Is it possible to calculate the volume of a parabolic arch?

Given that you know the equation of a parabola that only has positive values, is it possible to find the volume of the parabolic arch itself? NOT the volume of space underneath the arch. I asking ...
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1answer
22 views

Find the area of triangle APB, where P is a point $(a\cos\theta, b\sin\theta)$ on an ellipse and $A, B$ are its radii points $(a,0) (0,b)$

A point $P(a\cos\theta, b\sin\theta)$ sits on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. The points $A$ and $B$ have coordinates $(a,0)$ and $(0,b)$ respectively. Show that the area of ...