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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40
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4answers
2k views

Do “Parabolic Trigonometric Functions” exist?

The parametric equation $$\begin{align*} x(t) &= \cos t\\ y(t) &= \sin t \end{align*}$$ traces the unit circle centered at the origin ($x^2+y^2=1$). Similarly, $$\begin{align*} x(t) ...
18
votes
2answers
1k views

Intersection of two parabolae

Problem: Consider two parabolae such that their axes of symmetry form a right angle. Prove that all four points of intersection lie on a common circle (it is an assumption that there exist such four ...
3
votes
1answer
171 views

Introduction to parabolae with linear algebra

Linear algebra helps to introduce ellipses and hyberbolas. For example an ellipse can be seen as a transformed circle by a linear application. There is also this theorem for the curve ...
0
votes
1answer
247 views

How do I show that a parametric equation intersects the directrix?

The question was: The points P and Q on the curve: $$x = 2at, y= at^2$$ have parameters p and q respectively. Show that PQ intersects the directrix at: $$ \left (\frac{2a(pq-1)}{p+q},-a \right ) $$ ...
4
votes
6answers
3k views

Plot $|z - i| + |z + i| = 16$ on the complex plane

Plot $|z - i| + |z + i| = 16$ on the complex plane Conceptually I can see what is going on. I am going to be drawing the set of points who's combine distance between $i$ and $-i = 16$, which will ...
2
votes
2answers
210 views

Finding minimum of the modulus of a 2 variable function

How to find the minimum value of $$|f(x,y)|$$ where $f(x,y)$ is a 2nd degree function in x and y with no 'xy' term. $$f(x,y)=ax^2+by^2+cx+dy+e$$ How is the process different from finding the minimum ...
3
votes
1answer
384 views

Finding minimum of a two variable 2nd degree function under a certain constraint?

How to find the the minimum non-negative value of a function: $$f(x,y)=ax^2+by^2+cx+dy+e$$ s.t. $x$ lies in $[0, A]$ and $y$ lies in $[A, \infty),$ where $A$ is a known constant. or simply $0\leq ...
1
vote
2answers
4k views

Find the equation of an ellipse given its focus, directrix and eccentricity

Ellipse has a focus $(3;0)$, a directrix $x+y-1=0$ and an eccentricity of $1/2$. Find its equation. I should probably use the fact that $r/d = e$, where $r$ is the distance from the focus to any ...
1
vote
1answer
256 views

Trajectory Problem with Parabola

The height of an object is given by $$ h(x) = -0.005x^2 + x. $$ When does the object hit the ground? When does it attain its maximum height? What is its maximum height? I divided -.005/-1 to get ...
5
votes
1answer
368 views

What is the path equation that is created with the middle point of a fixed length line segment that touching both ends to an ellipse.

Ellipse equation is $(\frac{x}{a})^2+(\frac{y}{b})^2=1$ and the length of line segment is $2k$, if we move the line segment all around of the ellipse while touching both ends to the ellipse. What is ...
0
votes
2answers
555 views

Convert ellipse parameter from General parametric form to General polar form

I am facing problem to convert ellipse standard parameters. Everything I say here is refer to http://en.wikipedia.org/wiki/Ellipse I know what are the General parametric form parameter . Lets call ...
4
votes
2answers
178 views

How to decide that a curve segment is not an ellipse line segment?

Let me ask a question , given any short curve segment , how can you decide that it is not an ellipse line segment by a finite calculations? Thank you in advance.
1
vote
1answer
103 views

Are two ellipse arcs always almost identical if they have the same end points and the same center of ellipses?

Edit : This question is better to be ignored until the following related question will be discussed enough. This question relates to I know "almost identical " is not mathematics. But if you have ...
10
votes
6answers
3k views

How to find an ellipse , given 2 passing points and the tangents at them?

Please answer to a question , how to find an ellipse which passes the 2 given points and has the given tangents at them. And one related question is that the given condition can decide just one ...
6
votes
1answer
365 views

Conics in $\mathbb{A}^2$; Hartshorne, Exercise 3.1

I'm trying to solve Exercise 3.1 in Hartshorne's Algebraic Geometry: Show that any conic in $\mathbb{A}^2$ is isomorphic to $\mathbb{A}^1$ or $\mathbb{A}^1-\{0\}$. I know from a previous ...
1
vote
1answer
430 views

Calculate perimeter from parametric form with an ellipse?

Suppose I have a thing such as an ellipse: $$\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1$$ now we can define it so that $\frac{x}{a}=cos(\theta)$ and $\frac{y}{b}=sin(\theta)$. I ...
5
votes
2answers
11k views

What is the equation of an ellipse that is not aligned with the axis?

I have the an ellipse with its semi-minor axis length $x$, and semi major axis $4x$. However, it is oriented $45$ degrees from the axis (but is still centred at the origin). I want to do some work ...
11
votes
1answer
1k views

The Ellipse Problem - finding an ellipse inside a triangle

The problem statement is as follows: A triangle is dissected into six smaller triangles by its angle bisectors. Prove that the intersections of the angle bisectors of each of these smaller triangles ...
2
votes
1answer
704 views

Find intersection(s) between parametrized parabola and a line

I'm trying to find the value(s) of the parameter $t$ at the intersection point(s) between a 2D general parabola (as a parametric function of $t$) and a line whose equations can be derived from two ...
4
votes
2answers
1k views

Finding Coordinate along Ellipse Perimeter

Given an ellipse at (0, 0), with height "h" and width "w", what's the "x" coordinate along the perimeter for a given "y" coordinate?
0
votes
1answer
550 views

How to find a generic parabola through 3 arbitrary points in R^2?

Given $(a,b)$, $(c,d)$, and $(e,f)$ (assume non-collinear and $a\neq c$, $c\neq e$, and $a\neq e$), is there a generic way to find a parabolic function between the three?
0
votes
0answers
430 views

How to find intersections of hyperbolas by MATLAB

Let's have two hyperbolas given by equations: $$\mathbf{r}^T\cdot \mathbf A_1\cdot\mathbf r+\mathbf b_1^T\cdot\mathbf r+c_1=0$$ $$\mathbf{r}^T\cdot \mathbf A_2\cdot\mathbf r+\mathbf b_2^T\cdot\mathbf ...
4
votes
1answer
115 views

Ellipse: Name for the ratio $a/b$?

Given an ellipse with semi-major axis $a$ and semi-minor axis $b$, is there a "common" (or at least standard) name for either $\frac{a}{b}$ or $\frac{b}{a}$? I keep wanting to (informally) call it ...
4
votes
3answers
2k views

Calculate intersection of two ellipses

I used the equations found here to calculate the intersection points of two circles: (P3 is what I'm trying to get) Except, now I want to do the same with two ellipses. Calculating ...
1
vote
2answers
948 views

Finding & Plotting equation of hyperbola given foci, and difference in distances between them.

I have to plot the hyperbola (3 of them actually) in MATLAB, and so it'd be good if I could find some sort of general formula. The foci do not necessarily have to be on the axes (e.g. $(5,3)$ and ...
0
votes
1answer
726 views

How to calculate the X Y coordinates of an ellipse with only the X and Y radius length?

I have an ellipse where the radius of x-axis = 100 and y-axis = 30. I have 3 objects where I want to evenly distribute it along the ellipse. I have already done this for a circle where both axis' ...
2
votes
1answer
290 views

How to describe foсi of en ellipse inscribed in the triangle thru triangles angles points?

I was looking at Marden's theorem and could not help but wonder how foсi of en ellipse inscribed in the triangle can be described thru triangles angles points?
2
votes
3answers
118 views

How to find orignal equations of type $y=ax^2+bx+c$. given 3 coordinate points?

Ok, simple question, having trouble understanding this in school. So given a set of 3 points (xy-plane), such as (40,30) (60,28) (20,25) i have to find the equation of the parabola. I ...
2
votes
2answers
264 views

Conditions for intersection of parabolas?

What are the conditions for the existence of real solutions for the following equations: $$\begin{align} x^2&=a\cdot y+b\\ y^2&=c\cdot x+d\end{align}$$ where $a,b,c,d $ are real numbers. ...
3
votes
2answers
8k views

How do I find the equation of a tangent line to a curve?

I'm given $x^2+2x-4$ at $x=2$ and I have to find the tangent line to this curve at that point...
0
votes
1answer
652 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 ...
4
votes
3answers
2k views

How to get the limits of rotated ellipse?

The box that an ellipse fits is easily calculated if there are no rotation, or if the rotation is ${x*90^o}$ (where x is an integer) is easy. For a (major radius) and b (minor radius), it is : ...
9
votes
1answer
3k views

Calculating Distance of a Point from an Ellipse Border

I'm thinking about using oriented ellipses to represent curves (dents/bumps etc.) in my physics engine, and have a few questions about working with them: What methods are there to finding the ...
2
votes
1answer
317 views

Need help with the proof of conic section

Prove that the intersection of a plane and a object consist of one cone and one upside-down cone where the tip of cone meet is either degenerate conic or conic Also, idenify in what situation, the ...
1
vote
0answers
71 views

Some question concerning curve of second order

Let $$F(x,y)=ax^2+2bxy+cy^2+2dx+2ey+f,$$ $$\phi(x,y)=ax^2+2bxy+cy^2,$$ $x,y \in \mathbb{R}$. Assume that for some $x_0, y_0 \in \mathbb{R}$ and for some $\alpha, \beta \in \mathbb{R}$ such that ...
0
votes
1answer
136 views

Parabola with a variable starting point

I am trying to build an equation where I could start at (x,y) which are known and create a parabola from that starting point. I have no idea where it intercepts the X or Y. I know where I want the ...
2
votes
1answer
416 views

Properties of Parabola / Optimization

I've been working through some past papers for an exam which I am due to be sitting tomorrow. In the Conic Sections paper from a couple of years ago, the following question came up: The path of a ...
1
vote
1answer
3k views

how to find the parabola of a flying object [closed]

how can you find the parabola of a flying object without testing it? what variables do you need? I want to calculate the maximum hight and distance using a parabola. Is this possible? Any help will be ...
1
vote
4answers
246 views

Difficult equations to rewrite as ellipses

I have this equality that defines an elliptic boundary. I am trying to rewrite it in the form of the equation of an ellipse, but I am having trouble doing that. How would I go about rewriting this ...
6
votes
1answer
941 views

Relationship between ellipsoid radii and eigenvalues

I should start by saying that I haven't done algebra for very long time. I recently have some work related to algebra, so I need some help to speedup. I went through a theorem in the book stating the ...
1
vote
1answer
518 views

Center of gravity of an ellipse

I think the center of gravity of an ellipse is the intersection point of it's two radius. But I didn't see it anywhere, so I'm having some doubt about it. Am I right? Thanks to all.
3
votes
4answers
2k views

Is an ellipse a circle transformed by a simple formula?

Does any ellipse $E$ have a circle $C$ such that you can obtain $E$ by transforming $C$ by a simple formula $F$? In details , both $E$ and $C$ have the same center and the axes of $E$ are the XY axes. ...
0
votes
1answer
208 views

How to compute the cross point between an ellipse and a straight line?

Please let me know how to compute the possible cross points between an ellipse and a straight line. In details , I know the basic properties of the two shapes. So if the ellipse had its center at the ...
0
votes
2answers
183 views

Hyperbola property

I am posting the following question under homework category. I hope I will have very good answer from mathematicians about conic sections. I have seen closely the conic sections and their ...
15
votes
3answers
14k views

Check if a point is within an ellipse

I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane. How do I determine if a point $(x,y)$ is within the area bounded by ...
8
votes
3answers
891 views

an important property of an ellipse

Good morning everybody. I would like to know the proof of the following observation on the ellipse. A circle is drawn with the right latus rectum as diameter. Another circle is drawn with its ...
3
votes
3answers
291 views

Apostol Section 13.25 #13 - Conic Sections

Question: Prove that a similarity transformation (replacing $x$ by $tx$ and $y$ by $ty$) carries an ellipse with center at the origin into another ellipse with the same eccentricity. (The next ...
4
votes
2answers
338 views

Maximize the distance between a line normal to an ellipse and its center

My friend sent me this problem, which (upon Googling) seems to be from a Cornell class (1220?). Anywho. My advice to him was to parametrize the ellipse (say, in the first quadrant) with $x = a ...
-12
votes
1answer
776 views

A Hunt for a Mathematical Machine That Gives Points

The central question is : Is there any method for Producing the global Points on the curve (any cubic curve, or at least a Degree-2 curve ) , if we have local Part with us ? Explanation: ...
0
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2answers
330 views

why we only have a approximation for every circumference for ellipse but not define a special formula for each ellipse

Why do we only have an approximation for every circumference for ellipse, but we cannot define a special ratio formula for each ellipse? Is it possible for people to use a computer to find the exact ...