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
291 views

Locus of the centers of the circles tangent to a given line and circle

Say you are given a circle $C$ and a straight line $l$ exterior to the circle. How to describe the set of centers of circle that are tangent to both the $C$ and $l$? I have no idea how to proceed. My ...
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1answer
147 views

How to find the length of the focal chord making angle $\theta$ with the axis of parabola?

A focal chord of $Y^2 = 4aX$ makes angle $\theta$ with the axis of the parabola. How can I find the length of the chord? I have used the parametric equation but couldn't go further.
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2answers
282 views

How to derive the equation of a parabola from the directrix and focus

Could someone please offer me proof and explanation of the following? - I am just having trouble with finding the '$a$' part of the equation. "The leading coefficient '$a$' in the equation $$y−y_1 =​...
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0answers
102 views

Fitting an ellipse such that the ratio of its radii is in a range

I need to fit an ellipse to a group of points. However, I have an issue and I appreciate if anyone can help me. The issue is that I need to have the fitted ellipse such that the ratio of its radii is ...
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1answer
51 views

Difficulties in understanding ellipse's minor axis's equation

I'm implementing an ellipse detector using some pdf I found on the internet, but I encounter some difficulties in understanding one of the equations. Here is the pdf: http://hci.iwr.uni-heidelberg.de/...
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0answers
245 views

Finding equation for conic section given five points

Problem: Given the points $$(0,1),(0,-1),(2,0),(-2,0),(1,1)$$ find the equation for the conic section that passes through these points. My attempt: Using the general equation for a conic section, $$...
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1answer
19 views

Reduction of general conic

The given equation is - $$3x^2 + 2xy + 3y^2 - 32y +92=0$$ To get rid of xy term i used the substitutions - $$x=p+q , y=q-p$$ Then the equation becomes - $$(p-4)^2 + 2(q-2)^2=1$$ which is an ellipse ...
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3answers
344 views

Construction of an ellipse

Is it possible to construct an ellipse with a line, compasses and a pencil? If yes, how and why is the construction correct?
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0answers
166 views

Finding the equation of a rational function or a conic section given three points

I have a rational equation derived from 2 points, $(2, 2)$ and $(10, 10)$. Solving for the rational equation gives the equation $$y = \frac{20}{12-x}.$$ What I want to happen right now is that given ...
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1answer
192 views

Super conic sections?

I know graphs of the form $A x^2 + B xy + C y^2 + D x + E y + F = 0$ are conic sections. But what would happen if I changed the highest power to 3? Would this be a new 3D shape, a 4D version of it, or ...
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1answer
378 views

Why are elliptic/parabolic/hyperbolic PDEs called “elliptic”/“parabolic”/“hyperbolic”?

I see that the form of a (e.g.) parabolic equation is $$Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0$$ with $B^2-4AC=0$ whereas the equation of a parabola is $$Ax^2 + 2Bxy + Cuy^2 + Dx + Ey + ...
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1answer
120 views

Sections of cones in higher dimensions

Everybody knows that when a plane intersects a cone at different angles and positions, we get conic sections. But, I wanted to know that if the same was possible in higher dimensions. If we take the 4 ...
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2answers
94 views

Conic sections directrix and focus

I do not understand the following: The equation of a particular parabola is: $$(y−​23)​​ = -\dfrac{1}{​16}\!\!\!\!\!​​​​(x+3)​^2​​$$ Given the equation of a parabola is - $(y−y_1)​= a(x−x​_1\!\!)^{​...
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2answers
81 views

hyperbolic tangent vs tangent

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side (in a right triangle). Is there a similar definition for the hyperbolic tangent? The reason ...
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2answers
26 views

Show that the surface $x^2+y^2=x$ using $\theta \space and \space z$ can be parametrised by $(\cos^2(\theta), \cos(\theta) \sin(\theta), z)$

I really have no idea how to do this: $x^2-x+y^2=0$ looks like it can be a circle given by: $(x-\frac{1}{2})^2+y^2=\frac{3}{4}$ mostly $x=r\cos(\theta) \space and \space y=r\sin(\theta)$ work as ...
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1answer
204 views

Hyperbola / Rotated Hyperbola Intersection

I am trying to find the point where two hyperbolas intersect, that is, to find a vertex that is common to both hyperbolas. Also, note that I am only testing for a region of both hyperbolas -- only a ...
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4answers
88 views

Standard Form for a Parabola

What is the standard form for the following problem? I already know that it is a horizontal parabola. I just can't seem to be able to change it into the standard format. $8y² +96y-12x+240 = 0$ I ...
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1answer
37 views

Standard Form of Hyperbolas

If I have the equation $9x^2-4y^2-72x=0 $ and I know that is a hyperbola, how would I find the standard form for this equation? I'm not sure how to convert this equation to the standard form of a ...
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1answer
50 views

Conic Sections and Foci of Ellipses

We're just learning about ellipses and conics, and I'm a bit confused with ellipses, parabolas, circles, and hyperbolas, so a little help with this sample problem would be great. In which of the ...
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1answer
67 views

Partial Integral of an ellipse

this is my first question on stack exchange so please bear with me. I am trying to generate a synthetic image of an ellipse in Matlab where each pixel is shaded according to how much of that pixel ...
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0answers
68 views

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
28 views

Non-standard 3D rotation of a set of points [duplicate]

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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1answer
31 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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1answer
71 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
233 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 $$ ...
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1answer
337 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 ...
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1answer
64 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
52 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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6answers
1k 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
29 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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2answers
140 views

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
103 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
89 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
132 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
126 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
260 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 conic, ...
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3answers
268 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
425 views

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 ...
3
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1answer
83 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: $$ \frac{(y_4-y_1)}{(x_4-...
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1answer
469 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
35 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
117 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 I'...
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2answers
68 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 (y-11/30)^...
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0answers
86 views

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
85 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 vertical)...
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1answer
4k 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.
2
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2answers
107 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
109 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
64 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
87 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 :(