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
43 views

Longest parallel chord of an ellipse

I am searching for a source demonstrating that, for any set of parallel chords spanning an ellipse, the longest chord passes through the center of the ellipse. I am not referring to the major and ...
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1answer
29 views

Moving between different ellipse representations

I have a representation of an ellipse that is the affine transform of the unit ball, $\|Ax + b\| <= 1$. My question is, how can I change this ellipse representation? I would like to have it in ...
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1answer
69 views

What software should I use to graph this? / How do I get rearrange this equation so that it is in terms of y?

I thought I'd just quickly tell you guys why I want to graph this equation before giving it you. We're studying conic sections at the moment, and I started wondering what would happen if I let the ...
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1answer
21 views

How do I find the width of a given section of an ellipse?

How would I be able to find the width of a horizontal ellipse (with a major axis of 120 and a minor axis of 5) at any given point along the major axis?
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1answer
65 views

Find the arc-length of the circle with radius a?

Find the arc-length of a circle with radius a. From the equation of a circle, I found out the equation for the one quadrant, which is: $y = \sqrt{a^2 - x^2}$ I tried solving the problem, and here's ...
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0answers
83 views

Conic equation from cone/plane intersection

In an orthonormal cartesian frame $(O; \vec{x}, \vec{y}, \vec{z})$ consider: an infinite plane $P$ defined by: a point $p = (p_x, p_y, pz)$ an normal vector $\vec{n} = (n_x, n_y, n_z)$ a cone $C$ ...
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1answer
137 views

What are the coordinates of the ends of the latus rectum of the parabola $x^2 - 2y + 2 = 0$? [duplicate]

I've already graphed the parabola . i just don't know how to locate it's focus and the ends of it's latus rectum. On my graph, the vertex is on (0,1). Please help me with this. ASAP.
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34 views

Show that endpoint of a focal chord is $$\left(\frac{4p^2}{x_0}, \frac{p^2}{y_0}\right)$$

If $PQ$ is a focal chord of the parabola $x^2=4py$ and the coordinates of $P$ are $(x_{0}, y_{0})$ show that the coordinates of $Q$ are $$\left(\frac{4p^2}{x_0}, \frac{p^2}{y_0}\right)$$ I labeled ...
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1answer
111 views

Minimum eccentricity of ellipses around another ellipse

Six circles can surround another circle of equal size, with each circle touching both the central circle and its two neighbouring outer circles. For sufficiently eccentric ellipses, it is possible to ...
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1answer
36 views

ratio of tangent to the ellipse

The tangent at point $P = ( a \cos \phi, b \sin \phi)$ on the ellipse $\frac{x^2} {a^2} + \frac{y^2}{b^2}=1$ meets the $x$ and $y$ axes at the points $X$ and $Y$, respectively. Find in terms of ...
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2answers
158 views

Finding the maximum and minimum values on ellipse [closed]

Find the maximum and minimum values of f(x, y) = 5x + y on the ellipse x^{2} + 4y^{2} = 1
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1answer
301 views

Ellipse Word Problem

The ellipse is 5 meters across and 8 meters long with decorative fountains located at the foci. How far from the center should the fountains be located? (Rounded to the nearest hundredth). How far ...
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4answers
97 views

Find the equation of a circle passing three points (conics)

Problem: Determine the equation of the circle that passes through three points, $J(-3, 2)$, $K(4, 1)$, and $L(6, 5)$. I thought of using systems like so: $$\left\{ \begin{array}{rcl} (x+3)^2 + ...
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2answers
62 views

Conic Sections Question - Hyperbolas & Circles

So, if you have a hyperbola with foci at $(4,0)$ & $(-2,0)$, and the slopes of the asymptotes are $+4$ and $-4$, what would the equation for this hyperbola be? I know that the center would be ...
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1answer
43 views

How do I show that the equation E(k) = 2-4cos(ka) is a parabola when k=0 and when k=pi/a?

It's evident from the graph but I'm not sure how to show this mathematically. This dispersion relation is supposed to be roughly parabolic
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2answers
94 views

Conics: Asymmetric Hyperbola

I'm sure we've all seen the image below that illustrates the creation of the four conic sections. Although I've seen this multiple times throughout my education, I find it odd that the following case ...
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1answer
33 views

Intersection between conic and line in homogeneous space

In homogeneous space (so 3 coordinates for each point) I have: A conic C, defined by a symmetric 3x3 matrix of real values. The conic actually should have only imaginary points (don't know if this ...
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0answers
41 views

any conic in $\mathbb{A}^2$

Exercise 3.1 in Hartshorne's Algebraic Geometry: Show that any conic in $\mathbb{A}^2$ is isomorphic to $\mathbb{A}^1$ or $\mathbb{A}^1-\{0\}$. when the conic given by $x^{2}+y^{2}-1$, what the ...
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1answer
64 views

Non-linear systems help!

I have a non-linear system of equations, $$\left\{ \begin{array}{rcl} x^2 - xy + 8 = 0 \\ x^2 - 8x + y = 0 \\ \end{array} \right.$$ I have tried equating the expressions (because both equal 0), which ...
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1answer
89 views

Show that lines created by certain points on the parabola intersect at the directrix?

Edit: I got the answer by finding points of intersection between the line passing through B and the focus and the parabola, but it didn't seem like the best solution. Any other ideas? The Segments ...
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0answers
107 views

Find points of intersection with cone on a plane at a given angles

The provided variables are the cone angle(cA) of a cone that starts at the origin along the Z axis, the vertical angle (vA) of the direction the cone is facing, and a horizontal angle (hA) along with ...
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1answer
37 views

How to compute characteristic polynominal of two conics

If I have two conics defined as $A: XAX^T$ and $B: XBX^T$ how can I expand characteristic polynomial $f(\lambda) = det(\lambda A + B) $ so that it can be computed by a computer program or Matlab?
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1answer
43 views

How to find the height ($z$) on an elliptic cone at a point $(x, y)$

I am attempting to write a java method which returns the height of an elliptic cone given a $(x, y)$ point within the base. I have an elliptic cone centred at $(x_1, y_1)$, the major axis a, minor ...
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0answers
72 views

Finding a positive definite matrix to satisfy the general equation of an ellipse

I am trying to find a matrix A such that $(1)$ can be written as $v^TAv=1$ where $v=(x, y)^T$. $(1)$: $$\left(\frac{x}{a_1}\right)^2 + \left(\frac{y}{a_2}\right)^2 - ...
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1answer
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Problem in conics question

A vertical line passing through the point ($h$,0) intersects the ellipse $$\frac{x^2}{4}+\frac{y^2}{3}=1$$ at the points P & Q.Let the tangents to ellipse at P & Q meet at the point R.If ...
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2answers
66 views

Area inside an ellipse

Given the ellipse x^2/25\ + y^2/16 = 1, A = (5,0), B = (0,4); Find point C (with both coordinates positive) on the ellipse, such that the area between AC and the ellipse (S1) will be equal to the area ...
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1answer
64 views

Is this a correct way to derive the equation of an ellipse/hyperbola?

I was just testing to see if I could derive the equation of an ellipse (and consequently a hyperbola) with the least amount of information to remember. The small amount of information I chose to use ...
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1answer
99 views

Ellipse Tangents in 3D

I know that we can find the tangent of the ellipse in 2D by taking the derivative of the equation defining the ellipse. But I'm little bit confused about finding the ellipse tangent in 3D. Where the ...
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1answer
74 views

Homography between ellipses

This is a spin-off from a comment on Stack Overflow. How can I find a homography between two ellipses in the plane?
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26 views

How are the sine functions along with the hyperbolic functions visualized with imaginary rotations?

Since we know that: cos(t)=cosh(it) and isin(t)=sinh(it) I've been thinking about this, and obviously this is referring to how if you move at a right angle from a circle on a conic section, you end ...
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2answers
157 views

Area under parabola using geometry

We have to find the area of the pink region. As we all know this can be evaluated using limiting its Riemann sum, of which its a standard example. However I want to know if this can be done without ...
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1answer
38 views

to divide quarter of an ellipse into two equal halves

I wanted to divide the quarter of an ellipse into two equal halves. In what angle should I divide it so that both the arcs formed are equal in length. Finally I wanted to find the midpoint of the arc ...
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1answer
50 views

How to find position of object on a parabola?

I'm making a computer game, where an object flies along a parabola curve. This object is 'thrown' by a 'robot' towards another robot. I know the vertex of the parabola, and also it's two ...
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1answer
58 views

Finding vertex of a parabola -conflicting answers

For a given problem $g(x) = 5x^2 - 2x +1$, we were expected to describe the graph. WolframAlpha and I are finding conflicting issues. I do not think I made an approximation anywhere that would explain ...
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0answers
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History behind the choice of letters $h$ and $k$ for the vertex of a parabola?

After failing to find a historical explanation for usage of letters $h$ and $k$ for the vertex of a parabola in most relatively recent textbooks in anglosphere, I turn to math.SE. Is there any ...
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1answer
67 views

Concurrency-Three parabolas sharing common directrix.

I have found this result by exploring for new problems. If three parabolas share a common directrix and each pair intersect each other in two points, then, the lines joining the two intersection ...
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2answers
59 views

Find the expression of a parabola. [closed]

I need to figure out the expression of this parabola with the points $(-2,2)$, $(0,1)$, and $(1,-2.5)$.
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3answers
91 views

General form of a circle

My math teacher taught me that the general form (equation) of a circle is: $$ ax^2+by^2+cx+dy+e=0 $$ He also asked us this: If the product of $c$ and $d$ is negative, then what 2 quadrants can the ...
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5answers
107 views

The equation of parabolas.

I have trouble grasping some basic things about parabolas. (This should be easily found on Google, but for some reason I couldn't find an answer that helped me). I know one simple standard equation ...
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1answer
48 views

Calculating a position of an object on a parabola.

I am working on a simple 2D computer game. In the game, I have a 'robot' that throws a ball towards another robot, in the shape of a parabola. Both 'robots' are positioned on the x axis, aka their y ...
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1answer
46 views

Determining center of Ellipse with limited Data Points

The dataset I am using only has 200 degrees of the ellipse. The ellipse is not centered at (0,0). The data in this case ranges ...
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1answer
134 views

Calculate Ellipse From Points?

How can I calculate an ellipse from a group of points ? Result: center point, x-radius, y-radius ? I'm not mathematician so I don't really know the best parameter style for ellipses. This ellipse ...
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1answer
72 views

Intersection of conics

By conic we understand a conic on the projective plane $\mathbb{P}_2=\mathbb{P}(V)$, where $V$ is $3$-dimensional. I'd like to ask how to find the number of points in the intersection of two given ...
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Find the eccentricity of the conic $4(2y-x-3)^2 -9(2x+y-1)^2=80$

Find the eccentricity of the conic $4(2y-x-3)^2 -9(2x+y-1)^2=80$ Solution : $4(2y-x-3)^2 = 4x^2-16xy+24x+16y^2-48y+36$ and $9(2x+y-1)^2 = 36x^2+36xy-36x+9y^2-18y+9$ $\therefore 4(2y-x-3)^2 ...
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4answers
157 views

Find $z$ such that $|z+1|+ |z-1|=4$

I have this problem: Find all points of the complex plane wich satisfy: $$|z+1| + |z-1| = 4 $$ I know this is an ellipse with foci 1 and -1, and i know the answer is : $$3 x^2+4 y^2 \leq 12$$ but ...
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1answer
71 views

Ellipse and circle

if $\alpha$, $\beta$, $\gamma$, $\delta$ be the eccentric angles of four points of intersection of the ellipse and any circle,prove that $\alpha+\delta+\beta+\gamma$ is an even multiple of $\pi$ ...
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1answer
88 views

Proving properties of an ellipse

I'm studying about ellipse and its properties. My reference is the following pdf: http://nebula.deanza.edu/~bloom/math43/ellipse-derivation.pdf My questions are from the very first page of the ...
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2answers
61 views

How to get radius at any specific point in ellipse

How to find radius of ellipse at any point $(x_1,y_1)$. We know semi-major axis and semi-minor axis i.e. $a$ & $b$. center of ellipse $(x_0,y_0)$. Somewhere I found. $$ r = \frac{ab}{\sqrt{ ...
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1answer
90 views

How to show that this line touches the hyperbola?

The question is: $PQ$ is a chord joining the points $\phi_1$ and $\phi_2$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If $\phi_1\,+\,\phi_2 = 2\alpha$, where $\alpha$ is constant, prove ...
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1answer
68 views

finding eccentricity of ellipse??

If the tangent at any point of the ellipse make an angle α with major axis and an angle β with focal radius of the point of contact then show that the eccentricity of the ellipse is given by ...