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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4
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5answers
158 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 ...
0
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2answers
26 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, ...
2
votes
2answers
125 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? ...
0
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0answers
60 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, ...
1
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0answers
47 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 ...
1
vote
1answer
75 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 ...
1
vote
1answer
69 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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votes
2answers
88 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 ...
1
vote
3answers
73 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 ...
0
votes
2answers
91 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
votes
1answer
59 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: $$ ...
1
vote
1answer
77 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 ...
0
votes
0answers
24 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 ...
1
vote
0answers
24 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 ...
0
votes
2answers
42 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 ...
1
vote
2answers
62 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 ...
2
votes
0answers
66 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 ...
1
vote
1answer
32 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 ...
1
vote
1answer
207 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.
0
votes
0answers
47 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 ...
2
votes
2answers
95 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 ...
0
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2answers
47 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
votes
1answer
50 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 = ...
0
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1answer
28 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 :(
1
vote
2answers
97 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 - ...
1
vote
0answers
32 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 ...
1
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0answers
32 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 ...
0
votes
2answers
90 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 ...
2
votes
2answers
55 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: ...
0
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0answers
47 views

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 ...
1
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1answer
38 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 ...
0
votes
4answers
184 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 ...
1
vote
1answer
45 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 ...
2
votes
1answer
57 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 ...
0
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1answer
26 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 ...
18
votes
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 ...
0
votes
1answer
21 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 ...
0
votes
1answer
25 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 ...
1
vote
1answer
55 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 ...
0
votes
1answer
78 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 ...
0
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1answer
36 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 ...
4
votes
2answers
55 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 : ...
2
votes
1answer
65 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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1answer
47 views

Conic involving circle question.

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
31 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 ...
3
votes
5answers
104 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 $x^2 =-2; y^2=1, 2xy =\frac{-1}{2}$ ...
3
votes
3answers
113 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 ...
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vote
3answers
46 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?
0
votes
1answer
89 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 ...
1
vote
1answer
39 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 ...