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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2
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1answer
334 views

Volume of ellipsoid bounded by two planes.

I need to find the volume of ellipsoid: $$5x^2 + {y^2\over25} + {3z^2\over4} = 1$$ if the ellipsoid is bounded by $z={-1\over2}$ and $z=1$ planes. I was able to find the total volume of the ellipsoid ...
0
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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?
1
vote
1answer
645 views

find the center of an ellipse given tangent point and angle

I have an ellipse with known major radius $r_x$ and minor radius $r_y$, aligned with the x- and y-axis. Given a tangent point $T$ and the tangent angle $\alpha$, how do I calculate the center $C$ ...
3
votes
3answers
979 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 ...
0
votes
1answer
52 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
35 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
5answers
52 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} = ...
3
votes
2answers
21 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 ...
48
votes
9answers
847 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
2answers
61 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 ...
-1
votes
1answer
27 views

Find the equation of the ellipse [closed]

This question is hard I really appreciate if anyone can solve it. Find the equation of the ellipse that has $e=1/2$, center in (0,0) and passes through $P(9/2,3)$.
0
votes
0answers
26 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?
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0answers
21 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
1answer
533 views

Maximum/Minimum of Curvature - Ellipse

Find the sum of the maximum and minimum of the curvature of the ellipse: $9(x-1)^2 + y^2 = 9$. Hint( Use the parametrization $x(t) = 1 + cos(t)$) Tried to use parametrization like that, but then ...
-1
votes
1answer
34 views

Parabola Application [closed]

A cannonball when configured to be fired at a certain angle would have a parabolic path with a maximum height of $50\mathrm{m}$ and a horizontal range of $20\mathrm{m}$. If the cannonball is placed at ...
-2
votes
1answer
19 views

Conic-Sections, Ellipse, Parametrics, Normals and Tangents [closed]

Find the equation of the tangent and normal to the ellipse defined parametrically by: $x=5\cos\theta$ and $y=3\sin\theta$ at the point where $\theta=\pi/4$.
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vote
0answers
11 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 ...
1
vote
1answer
845 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 ...
0
votes
1answer
36 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.
0
votes
1answer
990 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 ...
1
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0answers
21 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 ...
0
votes
1answer
735 views

Calculating Intersection of an Ellipse and a Line

I found this page which gave me some equations on solving the intersection of a line with an ellipse given a point on the line and the slope of the line: There Isn't much explanation but ...
0
votes
0answers
16 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 ...
0
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1answer
14 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 ...
0
votes
2answers
97 views

Find all natural number solutions to: $20x^2 + 11y^2 = 2011$

I believe that the equation $$20x^2 + 11y^2 = 2011$$ describes an ellipse. I don't know how to solve for the $x,y \in \mathbb{N}$ that satisfy this equation.
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0answers
22 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
30 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? ...
0
votes
1answer
992 views

Calculating center of the ellipse

How to find center of ellipse from two points (these are just points on the ellipse, not related to foci), and two radii ($r_x$ and $r_y$, from standard definition of the ellipse $\frac{x^2}{r_x^2} + ...
9
votes
4answers
9k views

Calculating a Point that lies on an Ellipse given an Angle

I need to find a point (A on this diagram) given the center point of the ellipse as well as an angle. I've been melting my brain all day (as well as searching through questions here) testing out ...
1
vote
2answers
400 views

Finding asymptotes of hyperbola

Given implicit function $F(x, y) = 0$, how can I find its asymptotes? EDIT: Sorry, my calculations were wrong. Here is correct function: $F(x,y)=\sqrt{(x-a)^2 + (y-b)^2} - \sqrt{(x-c)^2 + (y-d)^2} ...
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
50 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
19 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 ...
15
votes
5answers
3k views

Aunt and Uncle's fuel oil tank dip stick problem

This problem first came to me in high school, and a couple times since, and I even assigned it for extra credit in one of my calculus classes after I became a teacher. So I know the solution. What I ...
2
votes
0answers
41 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
64 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
488 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 ...
3
votes
4answers
2k views

Tangent of an ellipse to an outside point

Let $C$ be a curve that is given by the equation: $$ 2x^2 + y^2 = 1 $$ and let P be a point $(1,1)$, which lies outside of the curve. We want to find all lines that are tangent to $C$ and intersect ...
0
votes
1answer
23 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 ...
5
votes
3answers
12k views

What is the focal width of a parabola?

I'm not wondering what the formula is—I already know that. For a parabola in standard form of $(x-h)^2=4p(y-k)$ I know that the focal width is $|4p|$. But what does that mean, conceptually? What ...
0
votes
3answers
46 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
56 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 ...
2
votes
1answer
49 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 ...
5
votes
2answers
82 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 ...
0
votes
1answer
31 views

A parabola with a horizontal directrix has it's focus at (2,5). If the point (7,-7) is on the parabola:

A. Find a possible equation for the directeix. B. Using your results from A. Find the vertex. C. Write an equation for the parabola This question is giving me some major difficulty I have spent 3 ...
0
votes
1answer
23 views

Approximating the circumference of given ellipse

Say we got the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{24}=1$, and the goal is to find the circumference using line integrals. So I parametrized the curve by $x=5\cos(t)$, $y=\sqrt{24}\sin(t)$. By ...
0
votes
1answer
29 views

What do you get when you take a conic section in between a parabola and vertical?

The way conic sections are often described, if you take a section parallel to the double-cone, you get a parabola, and if you take a perfectly vertical section, you get a hyperbola. But what if you ...
1
vote
1answer
401 views

Finding the volume of a cone by integration of parabolic conic sections

I am working on a purely academic way of finding the volume of a right circular cone of height $h$ and radius $r$, (assume $h > r$), using integration of parabolic conic sections (conic sections ...
0
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2answers
29 views

Emulating a parabola in my game for a jump

I am currently having some trouble understanding how to plot a parabola with the x and y coordinates.In my game a player needs to jump from point a to point b and the jump would look something like ...
2
votes
3answers
64 views

Find the shortest distance between the point and a parabola

Find the shortest distance between the point $(p,0)$, where $p> 0$, and the parabola $y^2=4ax$, where $a>0$, in the different cases that arise according to the value of $p/a$. [You may wish ...