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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Probability Density Function for Randomly Oriented Ellipse

I have an ellipse with a long aspect of a and a short aspect of b. The equation for this ellipse is found on this post: What is the general equation of the ellipse that is not in the origin and ...
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0answers
10 views

Non-standard 3D rotation of a set of points

I want to create a 3D surface as shown in the figure below. Toward this, I thought if I rotate a set of points in $xy$-plane on a elliptical arc I may be able to get such a surface. I was thinking of ...
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0answers
27 views

Why is this conic dual problem infeasible?

The problem is: $$\min \ x_2 : Ax -b = [x_1 \ 2x_2 \ x_1]^T \ge_{L^3} 0$$ where $L^m$ is the Lorentz cone. Which I found to have an optimal solution when $x_2 = 0$. I have shown that the conic ...
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3answers
49 views

Find the equation of the circle passing through points - (5,7),(8,1) , and (1,3). [on hold]

Find the equation of the circle passing through points (5,7),(8,1), and (1,3). I need to use these general formula of circle. Please help me to solve this Now i got g=3/2 and f=-19/2
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1answer
23 views

What space curves can this theorem describe?

We were given the following theorem in our Vector Calculus class: THM: For space curve $R$ which does not pass through the origin, and which has a second derivative, the following are equivalent: ...
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0answers
25 views

If someone has a velocity of $32$ ft/sec, will they be able to ring the bell( more info below)? [on hold]

At a carnival, a new attraction allows contestants to jump off a springboard onto a platform to be launched vertically into the air. The object is to ring a bell located $20$ ft overhead. The ...
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0answers
25 views

How do you find the standard form of a parabola when you only know the vertex and the focus? [on hold]

Focus(9,0) Vertex(0,0) Please explain it step by step.
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1answer
11 views

Tangents of Rectangular hyperbola

P,Q,R are points on a rectangular hyperbola, and PQ perpendicular to PR. Prove that the tangent at P is perpendicular to QR.
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3answers
43 views

Generic rotation to remove Quadratic Cross-product

Show that if $b\neq 0$, then the cross-product term can be eliminated from the quadratic $ax^2 + 2bxy + cy^2$ by rotating the coordinate axes through an angle $\theta$ that satisfies the equation $$ ...
2
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1answer
38 views

Equivalence of geometric and algebraic definitions of conic sections

I have not been able to find a proof that the following definitions are equivalent anywhere, thought maybe someone could give me an idea: A parabola is defined geometrically as the intersection of a ...
1
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0answers
21 views

Find equidistant points along a mathematically known path/trajectory

Is there anyway to find the positions of points with equal distance from each other along a known shape? For example, I'd like to find equidistant points on an ellipse or a 2D eight curve the way ...
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1answer
16 views

Polar equations of circles and ellipses

I have been trying to convert some conic sections from rectangular to polar form. I am fine going the other direction (given polar, convert to rectangular), but am having trouble going the opposite ...
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1answer
26 views

How so I put these in Standard form? Circle, Ellipse or Hyperbola?

I need help putting these into standard form so I can graph them. Also need help figuring out which ones are which: $$25x^2-16y^2-150x+64y-239=0$$ $$9x^2+4y^2+54x-64y+301=0$$ $$x^2+y^2-6x+8y+3=0$$
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5answers
68 views

How to geometrically prove the focal property of ellipse?

How to prove geometrically that if we have a tangent of ellipse with focus F and F' in point P, that tangent is bisector of the angle between a line joining focus F to point P and the line F'P outside ...
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2answers
23 views

Explanation of graphical mathematical anomaly (for me, anyways)

I was working on some competition stuff when I came across the equation $y^2+2xy-x^2 = 0$, and the thing that surprised me was, when I graphed it, I got these two perpendicular lines at the origin, ...
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0answers
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Tangent line parallel to a chord in a parabola

In a parabola with equation $f(x) = ax^2+bx+c$, a chord $AB$, $A=(xa,f(xa))$ and $B = (xb,f(xb))$, is parallel to the tangent line at $x = \frac{xa+xb}{2}$. It's easy to verify that using ...
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1answer
69 views
+50

The intersection points are collinear

It is given a hexagon inscribed in a conic section. I want to prove that the pairs of opposite site intersect at three points of the projective plane that are collinear. How could we do this? ...
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0answers
44 views

What's the standard form of the equation of a line of a slanted parabola?

I have been trying to figure out the general form of a slanted parabola, but what I've gotten doesn't look like it would be accurate:$$(x-h)^2+(y-k)^2=\dfrac{d}{\sqrt{h}}$$Where $(h,k)$ is the focus, ...
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0answers
25 views

Real world application of slanted conics (parabolae especially)

I am writing a report on slanted conics of the form $$(x-h)^2+(y-k)^2= \dfrac{d}{\sqrt h}$$ Where $(h, k)$ is the focus, and $d$ is the directrix. Are there any real world applications for slanted ...
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1answer
61 views

angular velocity around ellipse

If I have velocity at perihelion/apphelion, distance away from sun at perihelion/apphelion, and orbital period. How can I find the angular velocity function for earth and subsequently all the other ...
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1answer
55 views

Classifying complex conics up to isomorphism as quotient rings of $\mathbb{C}[x,y]$

This is a continuation of the question I asked here. The problem is now: Let $Q = ax^2 + bxy + cy^2 + dx + ey + f \in \mathbb{C}[x,y]$ be a general quadratic polynomial, that is, $a,b,c \not= 0$. ...
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2answers
63 views

Find the Locus of the Orthocenter

Vertices of a variable triangle are $$(3,4)\\ (5\cos\theta,5\sin\theta) \\ (5\sin\theta,-5\cos\theta) $$ where $\theta \in \mathbb R$. Given that the orthocenter of this triangle traces a ...
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3answers
54 views

Calculate tangent point on ellipse

I'm trying to find a tangent point on an ellipse. Trying a lot, using answers found a.o. on this site, but obviously doing something wrong as I'm not getting any good results. I've added a sketch, to ...
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2answers
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Help creating equation for parabola word problem?

The cables of a suspension bridge create a parabola. The towers are 600 feet apart and 80 feet tall. If the cable touches the road halfway between the towers, what is the height of the cable at a ...
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1answer
45 views

Ellipse inscribed angles

On wikipedia in German, we find relations about two angles inscribed on parable and on hyperbole. The 4 points of the parabola $y = ax^2 + bx + c $ has the following property: $$ ...
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1answer
35 views

Vertex Equation of an inverse quadratic function.

I'm working on a graphing web tool using JSXGraph, The user should be able to draw different functions. I was able to allow the user to draw quadratic functions by creating the vertex of the function ...
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0answers
20 views

Calculating the Section of an Ellipse

Hi, I have been trying to make an interactive SVG web page with JavaScript that allows you to edit an ellipse and will calculate the area of 'pie' section based on the angles and dimensions you ...
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0answers
18 views

Triangles with vertices on conics and their foci

Let $A$, $B$, and $C$ be the lengths of the three sides of a triangle. Let $α$, $β$, and $γ$ be the measures of the angles opposite those three sides respectively. Mollweide's formula tells us that ...
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2answers
20 views

Ceva, Desargues and Pascal's theorems for conics

I was told in class today that these three theorems are valid in projective geometry and with conic sections (I'm taking a modern geometry class) but I can't seem to find proofs anywhere online, and ...
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2answers
52 views

Indefinite integral with sector of ellipse

An ellipse is given by the following equation: $$ 152 x^2 - 300 x y + 150 y^2 - 42 x + 40 y + 3 = 0 $$ After solving for the midpoint we have: $$ 152 (x-1/2)^2 - 300 (x-1/2) (y-11/30) + 150 ...
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0answers
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Maximum product of lengths involving secant drawn to a parabola.

A chord is drawn from a point $P(1,t)$ to the parabola $y^2=4x$, which cuts the parabola at $A$ and $B$. If $PA\cdot PB=3|t|$, what is the maximum possible value of $|t|$? All I can infer is that the ...
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1answer
20 views

Proving a statement about ellipses and Dandelin spheres.

I have the Dandelin sphere construction. That is, I am given a vertical cylinder with radius $r$ and two spheres of radius $r$ are put inside of it. A plane (horizontal or otherwise, just not ...
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0answers
19 views

Find Tilted Parabola Equation given vertex and angle

How to find the parabola equation like the picture below, given the vertex $(x$$_o,y_o)$ and theta orientation? please help. thankyou.
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0answers
15 views

Approximation for uniform load on parabolic cable along its arc length

I am doing analysis for cable structures. Let's say that the cable stretches from point A to point B and carries a vertical ...
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2answers
92 views

I got this “parabolic” curve from a book but cannot find the right equation for it

The diagram below is taken from a book on Indian Stupa architecture. It says that the profile is a "parabolic" one. I have tried y=x^2 and varied the domains ox x and y but couldn't find the right ...
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2answers
33 views

Ellipse and rectangle

An ellipse, whose equation is ${x^2\over9} + {y^2\over4} = 1$, is inscribed within a rectangle whose sides are parallel with the coordinate axes. Another ellipse is circumscribing the rectangle and ...
2
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1answer
37 views

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 = ...
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1answer
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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
55 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
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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
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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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0answers
30 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
63 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
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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
9 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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2answers
46 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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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
27 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
54 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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1answer
34 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 ...