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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3
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0answers
33 views

Pair of tangents to a hyperbola

How do I find the joint equation of the pair of tangents drawn to the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ from an external point $(x_1, y_1)$. My book says that the answer is $SS' = ...
4
votes
3answers
245 views

Angle between two parabolas

I'm a little confused about a problem that asks me to find the angle between the two parabolas $$y^2=2px-p^2$$ and $$y^2=p^2-2px$$ at their intersection. I used implicit differentiation to find the ...
1
vote
2answers
21 views

Height of Line Segment on an Ellipsis

I'm trying to find the equation for getting the height of the black line I show in the image below. The end point of the black line is the intersection point between the width of the square below and ...
-2
votes
3answers
25 views

Finding the equation of a circle given 3 points without using elimination [on hold]

find the equation of the circle using points (-4,-4) (-3,1) (2,0) without using elimination.
0
votes
0answers
23 views

How to derive the relation $T=S_1$ for the equation of a chord of an ellipse whose midpoint is given? [on hold]

How to derive the relation $T=S_1$ for the equation of a chord of an ellipse whose midpoint is given as ( x1, y1 ) ? where T = x(x1)/aa + y(y1)/bb - 1 and S1 = (x1)(x1)/aa + (y1)(y1)/bb - 1 where 2a ...
1
vote
2answers
18 views

Create Ellipse From Eccentricity And Semi-Minor Axis

So I am given the eccentricity of an ellipse and the radius semi-minor axis as well as the center of the ellipse. So in the example below we know the center of the ellipse is at ( 0, 0 ) and the ...
1
vote
1answer
29 views

Polarity on a Hyperboloid of one sheet

Given a quadric $Q = \{v \in \mathbb{R}^n \mid \alpha(v,v) = 1\} \subset \mathbb{R}^n$, defined by a bilinear form $\alpha: \mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}^n$, and an affine subspace ...
1
vote
2answers
32 views

How to find the surface area of revolution of an ellipsoid from ellipse rotating about y-axis

Suppose the ellipse has equation $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$. I understand the way to obtain the surface area of the ellipsoid is to rotate the curve around y-axis and use surface of ...
-9
votes
1answer
45 views

Algebra 2 help! [closed]

Write the equation of a circle with the given center and radius. Center (-2,3); Radius 8
0
votes
1answer
19 views

Polar conversions of coordinates and parametric equations

Express the polar coordinates $P\left(6, -\dfrac{\pi}{4} \right)$ in Cartesian coordinates. $\displaystyle x=r\cos{(\theta)} ,\ y=r\sin{(\theta)} \implies x^2+y^2=r^2 \wedge \theta = ...
3
votes
1answer
81 views

reference on $\sqrt{ax}+\sqrt{by}=c$ as a parabola?

Does anyone have a reference on the equation $$\sqrt{ax}\,+\sqrt{by}=c\ ?$$ Clearing square roots and rearranging gives $$ax+by = \frac{(ax-by)^2+c^4}{2c^2}$$ This is the equation of a parabola, so ...
0
votes
0answers
17 views

Finding the eccentricity of ellipse when a line is a normal to the ellipse

Finding the eccentricity of an ellipse when a line joining the foot of the perpendiculars from a point of a known ellipse (having eccentricity e) at 2 perpendicular lines(example the x and y axes ...
0
votes
0answers
12 views

Modeling smoke cloud as expanding Gaussian / ellipse

I am making a simplified model of smoke coming from a train's smokestack. You can imagine that if you want an accurate model you have to think in 3D and use computational fluid dynamics and stochastic ...
1
vote
1answer
65 views

Finding the intersection points of common tangents on a pair of non-intersecting ellipses

I'm having some trouble with this, I don't know why but for some reason it is giving me a lot of trouble. Ultimately I intend to implement it into a program for modelling something, but I cannot even ...
2
votes
3answers
224 views

Parametric coordinates of parabola?

Can $(a(\sin(t))^2 , 2a \sin(t))$ be the parametric coordinates of the parabola $y^2 = 4ax$ ? I found that these coordinates satisfy the equation of the parabola but my friend says that although ...
0
votes
0answers
19 views

How to deduce this ellipsis equation?

I was looking in my physics book and it basically gave out the polar equation for an ellipse, as well as a few other definitions (which I assume are the definitions from which this ellipse equation is ...
0
votes
1answer
41 views

Parabola equation in Fortune algorithm for building Voronoi diagram

in DeBerg's "Algorithms and Applications", the part about Voronoi diagram, i have encountered the following formula for parabola arising in the beach line for a site point: $$\beta := y = ...
2
votes
0answers
36 views

Rotating a 3d-ellipse equation?

So I have very limited Linear Algebra knowledge, and I'm trying to program a computer graphics application in Android using OpenGL. I understand my design is not great, so if you have questions as to ...
4
votes
2answers
41 views

Area of a polygon inscribed into an ellipse

I have recently found a paper describing that the percentage area error of a polygon inscribed within a circle can be calculated using the following formula. The output of the algorithm is a set ...
5
votes
0answers
46 views

Keplerian orbits and closest approaches to Earth.

This question arose out of a discussion on Space.SE, but I think it will appeal to mathematicians more than astronomers: Let's consider a small astronomical object following an ideal elliptic ...
0
votes
3answers
17 views

Finding equation of parabola when focus and equations of two perpendicular tangents from any two points on the parabola are given

If the focus of a parabola and the equations of two perpendicular tangents at any two points $P$ and $Q$ on the parabola are given, can we find the equation of the given parabola? If not, what ...
-1
votes
1answer
29 views

The outer parallel (offset) curve of an ellipse [closed]

The inner edge of a track has equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. The track has uniform width $d$. What is the equation of the outer edge?
1
vote
2answers
60 views

How to get the foci / focus Hyperbola

How to find the lower and upper focus? Hyperbola I started with this $$ 9x^2 + 54x - y^2 + 10y + 81 = 0 $$ and broke it down to $$ \frac{9(x+3)^2}{25} - \frac{(y-5)^2}{25} = -1 $$ center = (-3,5) ...
0
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2answers
44 views

$2x^2-16x+28$ into standard form

I think I'm just doing something stupid here, because I know it's not hard. Here's what I did: $$y-28+{\_\_\_}=2x^2-16+{\_\_\_}$$ $$y-28+{\_\_\_}=2(x^2-8+{\_\_\_})$$ $$y-28+16= 2(x^2-8+16)$$ ...
3
votes
1answer
44 views

How to find ellipse circumference using 5 points?

I want to find ellipse circumference using 5 points. I have 5 point of an arc of the ellipse. To reach my goal i know that i have to do the following things: First, I have to find the general ...
0
votes
0answers
14 views

Parabolae intersection

when in some textbook they say "the right angled intersection of two parabolae", do they mean the intersection of their symmetry axes? If not what do they mean?
0
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1answer
56 views

Matrix representations of parabola.

Continuing the epic quest on finding matrix representations from here: Representation of hyperbolas. with a last part, the only conic section left: the parabola. I will present one idea of how to ...
4
votes
1answer
40 views

Start and end point of a rotated ellipse

I have the data of an incomplete ellipse and I need to retreive the minimun information in order to describe an elliptical arc. In particular following are my ellipse data: Major axis vector (x, y) ...
3
votes
2answers
24 views

The Reason for different Forms of Equations

I recently started learning about conic sections and saw people writing the equations for the different figures (circle, parabola, ellipse, and hyperbola) in different forms. (standard form, vertex ...
3
votes
5answers
53 views

Showing that certain points lie on an ellipse

I have the equation $$r(\phi) = \frac{es}{1-e \cos{\phi}}$$ with $e,s>0$, $e<1$ and want to show that the points $$ \begin{pmatrix}x(\phi)\\y(\phi)\end{pmatrix} = ...
0
votes
2answers
72 views

Determine the locus of a equation Quickly[Mental Math]

if $Z=X+iy$ then determine the locus of the equation $\left | 2Z-1 \right | = \left | Z-2 \right |$.I can tell that it a circle equation and it is $x^2 + y^2 = 1$.There are a lot of equation in my ...
0
votes
0answers
30 views

How to scale x- and y- axes equally in Maple?

I have the ellipse $\frac{25}{36}x^2+\frac{5}{36}y^2=1$. Maple draws it as a circle: How can I change the coordinates, to make it look like an actual ellipse?
1
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0answers
26 views

Plane and Ellipse Intersection

Short Version: If some can solve the easier to read form as follows, I would be thankful. \begin{equation} B = \frac{1 - d^{T}Bd}{ K_{1} } A \end{equation} \begin{equation} B^{T}d = \frac{1 - ...
1
vote
0answers
13 views

Projective conic generated by a set of tangent triangles.

I need to proof the following result: Let C be a real projective conic and P, Q two points interiors to C then there is another real projective conic such that every triangle inscribed on that conic ...
0
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1answer
37 views

Attempting to put the following conic into standard form: $y^2-2x^2+8y-8x-4=0$

Put the following conic into standard form: $y^2-2x^2+8y-8x-4=0$ I ended up with $$ -\frac {(x-2)^2}{12} + \frac{(y+8)^2}{24} = 1, $$ but I'm not sure if this is right.
1
vote
0answers
22 views

Strong duality in conic programmig

Let $K$ be a convex cone which is not closed. The look at a probem of the from $$\min <C,X>, \,s.t\; <A_i,X>=b_i,\, X\in C.$$ Now suppose I now that both this program and its dual have a ...
50
votes
9answers
954 views

Why does $\cos(x) + \cos(y) - \cos(x + y) = 0$ look like an ellipse?

The solution set of $\cos(x) + \cos(y) - \cos(x + y) = 0$ looks like an ellipse. Is it actually an ellipse, and if so, is there a way of writing down its equation (without any trig functions)? What ...
0
votes
1answer
15 views

Cone Plane Intersection Radius Size

I have a cone which is passing through a plane. The cone is not perpendicular to the plane, so the intersection area between the cone and the plane will not be circular but an ellipse. The cone will ...
1
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0answers
44 views

Differential Equations: Confocal Ellipse and Hyperbola

I am currently brushing up on Conic Sections, and I am having some problems on solving a first order quadratic differential equation. I would appreciate any help on the topic! I know that confocal ...
0
votes
2answers
37 views

Questions on the relation of the axis of a cone to its conic sections

(1) Does the axis of a cone pass through the foci of any its conic sections? Consider the image below: Is the intersection of the axis of cone and the ellipse the same as the focus of the ellipse? ...
2
votes
3answers
43 views

Is this equation a parabola or a hyperbola?

In a 1972 paper by Robert Merton, the following equation is derived: $$\sigma(\mu;A,B,C,D)=\sqrt{\frac{A \mu^2-2B\mu+C}{D}}$$ This is known as the Markowitz frontier in finance. When this is ...
2
votes
0answers
51 views

Is the following a conic section

All vectors are in $\mathbb{R}^3$ and only $\mathbf{r} = \left[ x; y; z \right]$ is unknown. My question is does the following system define a conic section in the $x-y$ plane and, if so, how can I ...
0
votes
1answer
22 views

Location of an arbitrary point of an ellipse

Given this ellipse equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $(a>b>0)$ and $c:=\sqrt{a^2-b^2}$ aswell as the focal points $F=(c,0)$ and $F'=(-c,0)$, why can we say without loss of ...
2
votes
0answers
48 views

Solving Kepler's Equation

I've been working on simulating orbits. I've found that, when solving Kepler's equation, $M = E - \varepsilon\sin{E}$, I'm unsure about the solution to use. For a true anomaly $< \pi$, using the ...
4
votes
2answers
80 views

Classification of conics in hyperbolic plane

How many different types of conics exist in hyperbolic plane? Euclidean geometry has three, of course. But when I was trying to find out results for the hyperbolic plane, the best thing I found ...
0
votes
1answer
38 views

Parametric equations - locus at midpoint

Consider the parametric equations $x=-2t^2$ and $y=4t$ The normal at any point, P, cuts the x-axis at Q. Find the Cartesian equation of the locus of the midpoint, M, of PQ. Can anyone help get me ...
0
votes
3answers
47 views

Equation of a parabola that passes thorught 2 point with know slopes

I want to be able to solve for the equation of this parabola. Known Points A(2,1) Slope @ A=1/2 B(7.25,2.5) Slope @ B=1/5 nothing else is known/given, The picture shows that parabola's Axis of ...
5
votes
0answers
59 views

Visualising 3rd degree equations

I know that general second degree curve, i.e. $ax^2 + by^2 + 2hxy + 2gx + 2fy + c=0$ gives us the equation of different cross sections of a cone. Similarly, what does a third degree* curve actually ...
0
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0answers
18 views

Relationship between polar angle and tangent angle of a conic section

I'm trying to define the relationship between the polar angle (or gradient) of a conic section and the tangent angle (or gradient) without resorting to x,y coordinates of the tangent point, i.e. given ...
5
votes
2answers
99 views

What's interesting in latus rectum?

I'm a maths teacher in Italian secondary school and I've been spending some time trying to construct "meaningful" problems about conic sections. I particularly like problems which focus on practical ...