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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Bounding box of an n-dimensional ellipsoid

I want to calculate the (axis-aligned) bounding box of an n-dimensional ellipsoid. The case of a (2-dimensional) ellipse is shown in following figure: Does anyone know a scientific paper about this ...
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32 views

Rational points of conics over $\mathbb{Q}$

I am starting to read lecture notes on basics of arithmetic geometry by A. V. Sutherland. In the second lecture, there is a procedure how to decide whether a conic over $\mathbb{Q}$ has a rational ...
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1answer
29 views

Ellipse fitting

I am not a mathematician and I don´t know much about it but i need help to fit an ellipse to a series of points and calculate its eccentricity. I have coordinates in the cartesian plane. I managed to ...
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1answer
70 views

Calculation method for a British Standard based on finding an ellipse given tangents?

This problem stems from a British Standard for staircases, of all things (BS 5395-1 2010, diagram B4 p.27), which prescribes the position of stairs when they turn a corner. I've tried to find an easy ...
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1answer
37 views

Condition on x-coordinate of a point such that three distinct normals can be drawn to a parabola

The set of points on the axis of the parabola $2{(x−1)^2+(y−1)^2}=(x+y)^2$ from which three distinct normals can be drawn is the set of points (h,k) lying on the axis of the parabola such that h>3/2? ...
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Condition on a & b so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by$ [duplicate]

Find the condition on a & b so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by$ I tried finding the joint equation of tangents to the ...
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3answers
145 views

Calculate centers of circles from their ellipse perspective.

Originally there are 4 circles in a plane and after perspective transform we get four conics. Now I know the equation of those ellipses. How could I get the origin of those four circles ? I know ...
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70 views

Is a symmetric positive definite matrix always diagonally dominant?

A Hermitian diagonally dominant matrix $A$ with real non-negative diagonal entries is positive semidefinite. Is it possible to have a Hermitian matrix be positive semidefinite/definite and not be ...
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8 views

Is there a better way of showing that this maximises the range up the ramp?

Consider a system where a projectile is shot up a ramp. Let the ramp be inclined at some angle $\alpha$ and the projectile is shot at some angle $\theta > \alpha$ with fixed velocity $V$. If we ...
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1answer
85 views

Sum of distances

We consider the ellipse $$\frac{x^2}{p^2}+\frac{y^2}{q^2}=1$$ where $p>q>0$. The eccentricity of the ellipse is $\epsilon =\sqrt{1-\frac{q^2}{p^2}}$, and the points $(\pm \epsilon p, 0)$ of the ...
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1answer
73 views

how to find the common chord to 2 parabolas

is there any method to find the common chord of 2 intersecting parabolas. i was told that eq. of common chord of 2 parabolas is S1-S2 where S1 and S2 are equations of the parabolas which does not ...
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56 views

Calculating X & Y coordinates of a point offset from an ellipse at a given polar angle and intersecting point

I am using the following equations to identify the x and y coordinates of a point on an ellipse at polar angle θ. $x=\pm\cfrac{ab}{\sqrt{b^2+a^2 (\tan^2\theta)}}$ $y=\pm\cfrac{ab}{\sqrt{a^2+b^2 ...
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2answers
46 views

Curvature of Ellipse

We all know that the curvature of a circle is defined by the equation $$k=\frac{1}{r}$$ What about ellipses? In terms of major axis $a$, minor axis $b$, $x$ and $y$, what is the curvature of an ...
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3answers
129 views

Convert this equation into the standard form of an ellipse

$$\frac{\left(\frac{xa^2}{a^2y^2+\ x^2}-p\right)^2}{a^2}+\left(\frac{ya^2}{a^2y^2+\ x^2}-q\right)^2=k^2$$ Could someone please convert this into standard form of equation ...
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45 views

If $p(x,y)$ is a irreducible quadratic polynomial, then there is a line not intersecting $p(x,y)=0$

Let $p(x,y)\in \mathbb{R}[x,y]$ be an irreducible polynomial and $deg(p)=2$. Then $p$ defines a conic $Q$ in $\mathbb{P}^2(\mathbb{R})$ as $$Q=\{[x,y,t]\in ...
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How to convert general equation of ellipse to a form analogous to standard form?

Sorry for the bad title , please edit it to something better if you can. I need a procedure to convert the general equation of ellipse - $$Ax^2 + By^2 + 2hxy + 2gx + 2fy + c = 0$$ into ...
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33 views

Plotting an ellipsoid using eigenvalue decomposition, how much should the axis tilt?

Given equation of an ellipsoid: $$E = \{x | x^TP^{-1}x \leq 1\}$$ My prof introduced a method where by you decompose this ellipsoid to its corresponding eigenvectors and eigenvalues so you could ...
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1answer
41 views

Drawing Ellipse from eigenvalue-eigenvector.

If I have two eigenvalue $\lambda_1$ and $\lambda_2$ and two associated normalized eigenvector $\mathbf e_1$ and $\mathbf e_2$ respectively, and I want to draw ellipse, How can I know which eigenvalue ...
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1answer
29 views

distance of the 'waist' of a caustic in a parabola

I just want to compute the distance of the 'waist' shown in the graphic. (Since I can't upload a graphic I'll show the link below) Normal families of a parabola I don't know how to call it bt ...
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45 views

How to find smallest enclosing ellipse when multiple lines are given ? This ellipse needs to intersect all those lines

I'm trying to find the smallest ellipse in terms of circumference. I suspect the smallest enclosing ellipse will intersect some lines in one single point. Given a line l: px + qy + r = 0 , L : {p q ...
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13 views

Conic Equation Simplification

Lets say that given the foci and directrix of a hyperbola we solve for the conic equation to be $$16x^2-16y^2=256$$ Is it possible to divide by $16$ and simplify to $$x^2-y^2=16$$ or is that "not ...
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3answers
41 views

Ellipse and tangents

Given the ellipse $x^2-xy+y^2=\frac{3}{4}$ I want to determine the points that their tangent is perpendicular to the $x'x$ axis and parallel to the $x'x$ axis. Solution The points that their tangent ...
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2answers
55 views

Is there a link between parabola and hyperbola?

I've merely seen the hyperbola defined as the "set of all points in a plane such that the difference of the distances from two fixed points (foci) is constant.". Like here: ...
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37 views

Which conic section does z represent?

If a,b,c,d are four complex numbers such that $c/d$ is real and ad-bc is not equal to 0.z is defined as $\dfrac{a+bt}{c+dt}$ where t is a real number.Which conic section does z represent? My ...
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1answer
50 views

Question about specific arclength over an ellipse problem

to see an image of what I'm talking about click this link: https://i.gyazo.com/909ccf0113fd26d21797f411a756ba1e.png In this image, arclength A is what we desire to be calculated. Point P is given and ...
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2answers
41 views

Is the discriminant of a second order equation related to the graph of $ax^2+bxy+cy^2+dx+ey+f=0$?

Is the discriminant of a second order equation related to the graph of $ax^2+bxy+cy^2+dx+ey+f=0$? Most people who took precalculus know that $ax^2+bxy+cy^2+dx+ey+f=0$ is the graph of: An ellipse ...
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1answer
222 views

Interesting property of triangle and ellipses formed on its edges

I need to prove following theorem. It seems to work, and it seems to be intuitive but I think I am missing skills in geometry proofs to prove that. Clearer version of theorem: For any triangle $ABC$ ...
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2answers
30 views

Deducing the focus of a parabolic mirror

I've been sitting on it for past three hours but I am unable to spot anything interesting. The first part was obvious but I can't answer the second question (b ii). How can I proof this statement k=a? ...
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67 views

Parabola Terminology

In Danish we call the two halves of a parabola that goes out to each side from the vertex branches like branches on a tree. Is there a name for them in English? Are they just called halves or maybe ...
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25 views

Is there a faster way of finding the family of orthogonally intersecting parabolas?

Say I have a parabola $y=ax-bx^2$ where $a,b>0$ and $y=cx-dx^2$ where $c,d>0$. I would like to find some sort of relationship relating $c$ and $d$ with $a$ and $b$ such that the two parabolas ...
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2answers
134 views

Find the equation of the parabola with focus (2;1) and vertex in the origin.

I have to solve a problem which says to find the equation of the parabola with focus in A(2;1) and vertex in the origin. Any suggestions are welcome.
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1answer
23 views

Rearrangement of Parabola

I am attempting to show that the following expression - $2 [(P)x^2 + (P+Q)x + P)$ can be rewrriten as $2 (x+1)(Px + Q$)... but I have come to no help. I did manage to get to $Px (x+1) + Q(x+1)$ ...
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2answers
60 views

A question about conic section (ellipse).

I am asked to solve the problem what is the center of the ellipse with vertex $V_1=(1,3)$ and focus $F_1=(1,0)$ and eccentricity $e=1/2$. My answer is due to the following analysis and computation: ...
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2answers
33 views

How to find eccentricity of the conic $3x^2+3y^2-2xy-2=0$?

How to find eccentricity of the conic $3x^2+3y^2-2xy-2=0$ ? How to deal with the xy term cant understand.Help! Why isnt the standard formula working here for eccentricity of an ellipse which states ...
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1answer
31 views

How would you find the eccentricity of this conic section?

$4x^2 - 5y^2 - 16x - 50y + 71 = 0$ Thank you!
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2answers
52 views

Question involving gradient of a function.

We are given any arbitrary ellipse with focii $F1$ and $F2$ , $T$ is the unit tangent to the ellipse through a point $P$. Let $f$ be the sum of the distances of the of $F1$ and $F2$ from $P$ , we ...
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1answer
37 views

Fitting an ellipse to a point with the first and second derivatives specified

I am trying to fit an ellipse to the end of a curve, $y(x)$, such that first and second derivatives of the curve, $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$, are preserved at the contact point, $(x,y)$, ...
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1answer
568 views

Parabolas in sequences of digits from the Fibonacci sequence

In preperation for an exam, I was studying Haskell. Therefore I was solving an old assignment where you had to define the fibonacci series. After solving the task (see 1] for source code) and ...
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1answer
74 views

Getting the angle that is needed for covering a given distance on an ellipse's cirumference

In a small programming exercise I asked myself, I want to calculate various things about ellipses. The part I'm stuck with is the following: I want to calculate the angle that is needed cor covering a ...
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1answer
50 views

How to transform the quadratic form of an ellipse to a circle

Consider the ellipse $x^TPx\le a$. I would like to transform (the quadratic form of) this ellipse into a circle $y^T\begin{pmatrix}1&0\\0&1\end{pmatrix}y\le b$ via a coordinate transform ...
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1answer
64 views

Show that an infinite number of triangles can be inscribed in either of the parabolas $y^2=4ax$ and $x^2=4by$ whose sides touch the other parabola.

Show that an infinite number of triangles can be inscribed in either of the parabolas $y^2=4ax$ and $x^2=4by$ whose sides touch the other parabola. I tried to solve it but failed.Can someone please ...
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1answer
183 views

Prove that the locus of the poles of tangents to the parabola $y^2=4ax$ with respect to the circle $x^2+y^2-2ax=0$ is the circle $x^2+y^2-ax=0$.

Prove that the locus of the poles of tangents to the parabola $y^2=4ax$ with respect to the circle $x^2+y^2-2ax=0$ is the circle $x^2+y^2-ax=0$. I have encountered this question from SL Loney.I have ...
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2answers
36 views

What is wrong with this formula?

I'm trying to make a formula that converts an ellipse in general form to one in standard. My steps to derive it are as follows: $$ax^2+bx+cy^2+dx+e=0$$ Move e to the other side... ...
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1answer
52 views

The equation of the circle ,having double contact with the ellipse at the ends of a latus rectum,is $x^2+y^2-2ae^3x=a^2(1-e^2-e^4)$

Prove that the equation of the circle ,having double contact with the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$(having eccentricity $e$) at the ends of a latus rectum,is ...
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1answer
44 views

If any two chords be drawn through two points on the major axis of an ellipse equidistant from the center

If any two chords be drawn through two points on the major axis of an ellipse equidistant from the center,show that ...
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1answer
46 views

Rotated parabola 2d vertex

I'm implementing an application where I need to get the vertex of a parabola, the parabola might be tilted; so it can have an angle with the x-axis not necessarily vertical or horizontal. Can I get ...
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4answers
128 views

Prove that the least intercept made on the tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ by the axes is $a+b$.

Prove that the least intercept made on the tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ by the axes is $a+b$.Also find the point of contact of the corresponding tangent. I tried.Let ...
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1answer
24 views

Clarification regarding ellipse

Is there only one ellipse possible with vertex as (1,0),any focus as (3,0) and eccentricity 2/3 ? Personally I feel there can be more than one situation like that.But wolfram alpha gives this ...
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1answer
31 views

Equation to get the center point of the union of n ellipses?

If I have 3 ellipses that all intersect such as in image. How can I get the center point of the Union of all three ellipses? (Basically the center point of the red area in the image)
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1answer
66 views

How to create quadratic equation given $y$ intercept, and maximum and $B=8$?

The given are Two x-intercepts y-intercept(0,-4) Maximum at (2,4) i tried everything i know...its been a long time since I have been doing math problems but the only way i thought about was to use ...