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
759 views

A hyperbola passing through integer lattice points

Prove that for any $n\geq 0$, there is a hyperbola that passes through exactly $n$ lattice points (= points with integer coordinates) and find an example. For example it is easy to see that the ...
3
votes
1answer
510 views

How do I get a tangent to a rotated ellipse in a given point?

I have just graduated from a school you would call High School and even though we talked about tangents to ellipses, we never covered rotated ellipses. So, what I am looking for, is a formula for a ...
3
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1answer
1k views

How many points are necessary to find a parallel ellipse, and how to do it?

So, I understand that to find an ellipse for sure you need at least five points. Why? The ellipse equation has only four variables ($x_0, y_0, a,\text{ and }b$). That's not actually my true question, ...
4
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3answers
6k views

Finding Asymptotes of Hyperbolas

To find a asymptote its either b2/a2 or a2/b2 depending on the way the equation is written. With the problem $$\frac{(x+1)^2}{16} - \frac{(y-2)^2}{9} = 1$$ The solutions the sheet I have is giving ...
2
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2answers
104 views

Ball from platform with specific vertex

On earth in a vacuum. You throw a platon from a platform height $h$ and want it to land at point $d$ distant. Note, h is absolutely fixed and d is absolutely fixed. It "must land" at point d, no ...
2
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1answer
209 views

How to find the eccentricity of this conic?

How to find the eccentricity of this conic? 4(2y-x-3)² - 9(2x+y-1)²=80 My approach : I rearranged the terms and by comparing it with general equation of 2nd degree, I found that its a hyperbola. ...
0
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2answers
128 views

Can someone help me with this conic?

$$\frac{(x+1)^2}{16} + \frac{(y-2)^2}{9} = 1.$$ I just started conics, but I thought you would multiply both sides by $16$ and then $9$ and then expand, which would get you $x^2 +y^2+2x-4y+5$. Both ...
3
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3answers
3k views

How to find an ellipse, given five points?

Is there a way to find the parameters $$A, B, \alpha, x_0, y_0$$ for the ellipse formula $$\frac{(x \cos\alpha+y\sin\alpha-x_0\cos\alpha-y_0\sin\alpha)^2}{A^2}+\frac{(-x ...
2
votes
1answer
390 views

Prove that a conic section is symmetrical with respect to its principal axis.

A Calculus book that I'm self-studying is asking me to prove the following theorem about conic sections: A conic section is symmetrical with respect to its principal axis. Here is my attempt at ...
3
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1answer
1k views

How to derive ellipse matrix for general ellipse in homogenous coordinates

So lets say we have an ellipse with axes a and b and the rotation angle $\phi$ and center at $(0,0)$. Now I apply the rotation to $x^2/a^2+y^2/b^2=1$ getting $$x' = x\cos(\phi) + y\sin(\phi)$$ $$y' = ...
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2answers
480 views

Get point on ellipse from point and angle

I have the bounds (x from, x to ...) of an ellipse (and thus its radius and center), x and y of a point A that is in the ellipse and an angle. I want to get point B to which the angle points (from ...
8
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3answers
591 views

If two parabolas don't intersect, is there a line that doesn't intersect either of them?

Two parabolas in a plane are given, such that they don't intersect. Is it true that there is a line in plane such that doesn't intersect any of them?
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2answers
6k views

Set up double integral of ellipse in polar coordinates?

How do you set up a double integral for an ellipse in polar coordinates without using Jacobian or Greens Theorem? I can't seem to figure out what (or if) the limits of r can possible be. $x = ...
3
votes
2answers
164 views

Integer points in a curve?

The question asks for necessary and sufficient conditions for a given curve to have integer points, that is, $(x_k,y_k)$ such that $x_k,y_k\in\mathbb{Z}$. For example, a necessary and sufficient ...
0
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1answer
204 views

Implicit equation of an arch in 3D

I have three points: A(85, 85, 0), B(-85, -85, 0) and C(0, 0, 30). I must find the equation of the arch that starts from A, finishes in B and goes through C. Could you help me? I found something ...
1
vote
1answer
3k views

Moment of inertia of an ellipse in 2D

I'm trying to compute the moment of inertia of a 2D ellipse about the z axis, centered on the origin, with major/minor axes aligned to the x and y axes. My best guess was to try to compute it as: ...
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2answers
173 views

Area of ellipse given foci?

Is it possible to get the area of an ellipse from the foci alone? Or do I need at least one point on the ellipse too?
1
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1answer
305 views

Marden’s Theorem

Given a triangle and three vertices in x, y format, is there a systematic way to use Marden’s Theorem to get the vertices of the foci of the inscribed ellipse? It seems to involve the derivative but ...
2
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2answers
347 views

Parabola Question: How to derive an appropriate equation based on specific criteria?

I've seen similar questions, but none worded quite like what I'm asking here. I'm not a math major, so I may have follow-up questions if answers befuddle me. Scenario: I'm trying to emulate a ...
4
votes
1answer
349 views

Problem in Silverman/Tate Rational Points on Elliptic Curves

I'm trying to figure out how to solve the following problem the "right" way. This is problem 1.2 on page 32: Let $C$ be the conic given by the equation $$F(x,y)=ax^2+bxy+cy^2+dx+ey+f = 0$$ ...
2
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1answer
186 views

Math function for parabola

I need an implicit function that plots the parabola that I am showing you in the picture. Everything you need is shown there. The radius of the thickness of the parabola must be 3. Thank you in ...
0
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2answers
170 views

How do I calculate the length of a vertical offset of the major axis in an elllipse?

Please forgive my terminology if it is imprecise. In the diagram below, for known values of X, Y and Z, I am need to calculate the value (length) of M. (It's not homework, it's for an SVG ...
0
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1answer
351 views

Finding a Parabola from its height and second y-intercept

I would like to know how to find the quadratic equation for a parabola that opens down and intersects the origin along with the vertex being in the first quadrant given the maximum of the parabola and ...
0
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2answers
231 views

Maximum area of 3 rectangles inside an ellipse

I'm trying to determine the maximum area in a specific ellipse that can be filled with any 3 (horizontally aligned) rectangles. $$Ellipse: \frac{x^2}{36}+\frac{y^2}{16}=1$$ $$Area: ...
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2answers
770 views

Calculating major axis of an ellipse

How do I calculate the length of the major axis of an ellipse? I have the eccentricity and the length of the semi-major axis.
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2answers
199 views

Find the parallels to a line which are tangent to an ellipse

Having the equation of a line, how can I find which of its parallels are tangent to an ellipse of equation $x^2 + 9y^2 = 1$? If the equation of the line is $y = mx + q$, I know that its parallels ...
0
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1answer
430 views

How to find the radius(major and minor) with the given 3 points in an ellipse?

I have 3 random points in an ellipse. Is it possible to find the radius of the ellipse?
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1answer
304 views

Intercept path to object following an elliptical path

I have a game with planet objects traveling in elliptical orbits. I want to plot a path from an object in one orbit to an object in another orbit. I'm not concerned with gravity etc as this is ...
3
votes
1answer
149 views

The probability of $Ax^2+Bxy+Cy^2 = 1$ defining an ellipse.

In Keith Kendig's paper, Stalking the Wild Ellipse (published in the American Mathematical Monthly, November 1995), he says that if $A, B, C$ are chosen at random, the probability that the Cartesian ...
3
votes
1answer
166 views

How to check if two 2nd degree conic curves intersect in a given region?

Let there be two 2nd degree curves: $$f(x,y)=ax^2+by^2+cx+dy+e=0$$ and $$g(x,y)=fx^2+gy^2+hx+iy+j=0,$$ how is it possible to determine if these two curves intersect in some region, say $x \le 1 , y ...
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0answers
750 views

Symmetry of a hyperbola?

What types of symmetry do all hyperbolas have? Do all hyperbolas have rotational symmetry as well as mirror symmetry?
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1answer
2k views

Ellipse in Quadratic Form: Finding Intercepts with Principal Axes

Where an ellipse is expressed in quadratic form (e.g. $ax^2 + bxy + cy^2 = k$ is expressed as $x^TQx = k$), the principal axes are in the directions given by the eigenvectors of Q. I understand this. ...
2
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3answers
274 views

Find the smallest $k\in \mathbb{Z}$ such that $f(x)=x^2-3x+k$ and $g(x)=x-2$ do not intersect.

$$ \large{ \text{Here are some instructions from the original question: } \ \\ } $$ $$ \large{ k \in \mathbb{Z} \ \ \land \\ f(x)=x^{2}-3x+k \ \ \land \ \ g(x)=x-2 \ \\ \text{And, there is NO any ...
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1answer
1k views

Eccentricity of an ellipse

How is $\frac{PF}{PD} = e = \frac{C}{A}$ ? where e is eccentricity, P stands for any point on the ellipse. $F$ stands for one of the foci. $e$ stands for eccentricity. $D$ is a point on the directrix ...
0
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1answer
91 views

Optimization Problem with Modulus function

Any ideas how to solve the following problem: $$Minimize: |F(x,y)|+|G(x,y)|$$ s.t. $x<A, y<B$ where $$F(x,y)=ax^2+by^2+cx+dy+e$$ $$G(x,y)=fx^2+gy^2+hx+iy+j$$ and $A,B$ are known constants. Any ...
0
votes
1answer
2k views

Find the intersection of a line (segment) and an ellipse (from the center of ellipse)

Here is what I know: The location of the center of the ellipse C (20,10). The Major axis (2a) or 400 (a being 200) - this is on the X axis. The Minor axis (2b) or 200 (b being 100) - on the y axis. ...
42
votes
4answers
3k 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
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2answers
2k 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
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1answer
174 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
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1answer
259 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: $$ \left (\frac{2a(pq-1)}{p+q},-a \right ) $$ ...
5
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6answers
4k 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
225 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
412 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 ...
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2answers
5k 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 ...
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1answer
299 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
383 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
625 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
181 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.
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1answer
106 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
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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 ...