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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17 views

multiple parabolas repeated hotizontally

I've been trying to write/find an equation which gives me ability to introduce dips on a parabolic graph on demand, basically, change a constant value or add a piece of equation to the original one to ...
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3answers
68 views

How to sketch $-3x^2 - 8xy + 3y^2 = 1$ [on hold]

The equation is as follows: $$-3x^2 - 8xy + 3y^2 = 1$$ How to specify the axis of the given curve? How to as accurately as possible draw a curve defined by this equation?
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1answer
476 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 ...
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2answers
30 views

What conic is $x^2-y^2-2y-2=0$ and what is the conic format of it? It would be great if work was shown so I could learn it.

What conic is $x^2 - y^2 - 2y - 2=0$ and what is the conic format of it? It would be great if work was shown so I could learn it. I know it's either a hyperbola or an eclipse but I don't know how to ...
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18 views

Fit an ellipse with known semi-major-axis and points

In my particular case I am given a projection of a circle onto the $xy$-plane and the radius $r$ of said circle. This results in an ellipse with semi-major axis $a$ equal to $r$. Like in this other ...
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2answers
31 views

How to solve this system of conics?

I am currently trying to figure out how to solve the following systems of conics: $\frac{(x+1)^2}{16} + \frac{(y-1)^2}{81} = 1$ $x+6=\frac{1}{4}(y-1)^2$ How would I find the four points that these ...
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1answer
29 views

Intersection of two parabolas where one is vertex shifted

I would like to be able to calculate the intersections of two parabola's which accounts for one or both of the parabola's being shifted along the x axis I have written an excel vba function to do ...
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4answers
57 views

How to find the common tangent to the curves $y^2=8x$ and $xy=-1$?

How to find the common tangent to the curves $y^2=8x$ and $xy=-1$ ? My approach: I used the formulae for tangents of a parabola and hyperbola.For any conic section if $y^2$ is replaced by ...
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1answer
87 views

An interesting property between a hyperbola & parabola

It is well known that when two tangents to a parabola are perpendicular to each other, they intersect on the directrix. In other words, the intersection point of the two tangents make a straight line, ...
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2answers
22 views

Test if a vector is pointing towards the center of an ellipse

I have an ellipse : $$x = h + a\cos t \cos\theta - b\sin t \sin\theta \\ y = k + b\sin t \cos\theta - a\cos t \sin\theta$$ Let's say if we have a normal vector $n$ to the ellipse, on a point $p$ ...
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1answer
37 views

vertices of a hyperbola the silliest question ever

I'm given that the center of the hyperbola is $(2,1)$ and $a=3$ and asked to find the vertices. Since vertices are on the same line with the axis of symmetry I thought the coordinates should be $(2,1 ...
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1answer
25 views

Axis angle and length of ellipse

For an ellipse defined by $$x = a \cos(t + \alpha)$$ $$y = b \cos(t + \beta)$$ What are the angles and lengths of each axis? I've tried to work backwards from the expression for a rotated ellipse ...
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1answer
27 views

Physical application of conics using a ladder

Hi so I've been given a question for a Maths assignment in relation to conics and its applications. The question is: A $6m$ ladder lies against a wall. Its bottom is pulled along the floor away from ...
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1answer
453 views

The reflective property of ellipses

I have following question to proof: An ellipse is revolved about its major axis to generate an ellipsoid. The inner surface of the ellipsoid is silvered to make a mirror. Show that a ray of light ...
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3answers
32 views

how to parameterize the ellipse $x^2 + xy + 3y^2 = 1$ with $\sin \theta$ and $\cos \theta$

I am trying draw the ellipse $x^2 + xy + 3y^2 = 1$ so I can draw it. Starting from the matrix: $$ \left[ \begin{array}{cc} 1 & \frac{1}{2} \\ \frac{1}{2} & 3 \end{array}\right]$$ I ...
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1answer
26 views

Elliptical section of a right circular cone [closed]

A right circular cone, having cone angle $\alpha=40^o$, is thoroughly cut with a smooth plane (normal to the plane of paper as shown by the produced line AB in the diagram below) making at an acute ...
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2answers
83 views

How to find center of a conic section from the equation?

If we are given a curve in the form $$ax^2+2bxy+cy^2+2dx+2ey+f=0$$ and the following determinant $$\delta=\begin{vmatrix}a&b\\b&c\end{vmatrix}=ac-b^2$$ is non-zero, then this is either a curve ...
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1answer
23 views

Need help with parametric parabola

so i was given my Math C assignment today and the moment i looked at question 1 i knew i had no idea what to do. This is the graph i was given (http://imgur.com/nRXOlJy). I was asked to provide an ...
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3answers
3k views

Calculate the intersection points of two ellipses

I used the equations found on Paul Bourke's "Circles and spheres" page to calculate the intersection points of two circles: $P_3$ is what I'm trying to get, except now I want to do the same with ...
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1answer
19 views

Conics - How to Prove

Not really sure how to approach part (iii) I have proved parts (i) and (ii), I'm assuming I have to use those answers. Any help would be greatly appreciated
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1answer
952 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 ...
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1answer
716 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 ...
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2answers
26 views

Tangents to ellipse from point outside curve

I was revising for one of my end of year maths exams, then I came across this example on how to find lines of tangents to ellipses outside the curve. Personally, I'd use differentiation and slopes to ...
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1answer
756 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 ...
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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 ...
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1answer
27 views

Finding angle of rotation of an ellipse

Suppose I have the ellipse $$ x^2 -2xy +4y^2 = 1 $$ How can I find the angle at which this ellipse is rotated? I have tried to assign $x=\cos\theta, y=0.5\sin\theta$ but I don't know if that's the ...
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1answer
21 views

Conic reduction

I'm trying to reduce this conic : $x^2+y^2+2xy+x+y=0$ to a canonical form. I started with finding the eigenvalues of the matrix associated to the quadratic form $x^2+y^2+2xy$ I found $z_1=2 , ...
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2answers
27 views

Finding the points of a line with a known direction and distance joining 2 ellipses

I have 2 ellipses, say $e_1$ and $e_2$. I want to draw a line $l$ connecting $e_1$ and $e_2$ in a known direction $(u,v)$, with a known distance $d$. Is there a way to solve for the points of ...
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1answer
25 views

Proving equations for conic sections?

How can we prove that the equations for conic sections are, indeed, sections of a cone? My guess is that it involves some sort of equality with the quadric surface equation for a cone, but I can't ...
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1answer
22 views

Whether or not a plane is a tangent plane to an ellipse or not? And if so, what is the point of intersection

Say we have an ellipse Transpose( p-c )A(p-c) = 1 and a plane x = a where Transpose (p-c) implies the transpose of the array p-c and a is a const A is an nxn matrix where the ellipse has ...
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2answers
39 views

Find the slope of line L [closed]

A straight line ($L$) passing through the point $A(1,2)$ meets the line $x+y=4$ at the point $B$. If $AB=\sqrt 2$, what is the slope of $L$? With some help I did it and the slope comes out to be 2+√3 ...
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1answer
23 views

Parabolas and lines…

Sooo... I have just received this question. 'Draw the graph of $y = x^2 + 3x - 2$'. Now, I can do this just fine. Then it says 'draw a line on the graph to solve the following equations. $x^2 + 3x ...
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0answers
33 views

Evaluate the eccentricity of the elliptical section of a right circular cone

A right circular cone, with the apex angle $\alpha=60^{o}$, is thoroughly cut with a smooth plane inclined at an acute angle $\theta=70^{o}$ with its geometrical axis to generate an elliptical section ...
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2answers
7k views

How do you find the distance between two points on a parabola

So I've been wanting to figure out a formula for an odd pattern I found... but to write a proof, I need to know one thing... How do I find the distance between two points on a parabola? Like, if I ...
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4answers
81 views

How does $x^2+4xy-6x+4y^2-12y+9=0$ represent a straight line.

I need to show $x^2+4xy-6x+4y^2-12y+9=0$ is a straight line. But I only know of a straight line in the form $y=mx+c$. Any help?
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0answers
12 views

$\text{SO}(2,1)$ is isomorphic to the group of automorphism of $x^2+y^2-z^2$

I am not able to prove the following. Let $\mathbb{K}$ be a field with more than four elements and with characteristic different from $2$. Let $\mathcal{C} \subset \mathbb{P}_2(\mathbb{K})$ be the ...
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1answer
472 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
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29 views

Curios relation between parabola, circumcircle and circumellipse

When playing around with conics in GeoGebra, I have found out that the following relation seems to hold: Let parabola $p$ be tangent to sides/extensions of sides $BC,CA,AB$ of triangle $ABC$ at ...
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0answers
29 views

Finding equation of line at a given angle from point to ellipse

Given a point $p_0$ and the parametric equation of an ellipse. I want to find the vector $v$ from $p_0$ such that when it intersects with the ellipse, it forms an angle $\theta$ with the ellipse's ...
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3answers
2k views

Are there parabolic and elliptical functions analogous to the circular and hyperbolic functions sin(h),cos(h), and tan(h)?

Are there parabolic and elliptical functions analogous to the circular and hyperbolic functions sin(h),cos(h), and tan(h)? Also, in matters of conic sections, are there other properties such that it ...
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1answer
64 views

Location of the foci of a hyperbola as the value of $a$ becomes increasingly smaller than the value of $b$

"What happens to the location of the foci of a hyperbola as the value of $a$ becomes increasingly smaller than the value of $b$?" I assumed that the hyperbola was in the form ...
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1answer
13 views

Conic Equations

I'm confused as to how you identify which equation for a conic is being used. For example, an ellipse has two equations, $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2}=1$ or $\frac{(y-k)^2}{a^2} + ...
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1answer
27 views

sum of the squares of the reciprocals of the two parts of the focal chord of a parabola

Find the sum of the squares of the reciprocals of the two parts of the focal chord of a parabola. My attempt: Let $y^2=4ax$ be a parabola. Let PQ be the focal chord through the focus S$(a, 0)$ ...
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2answers
2k views

How to convert the general form of ellipse equation to the standard form?

How to convert the general form of ellipse equation to the standard form? $$-x+2y+x^2+xy+y^2=0$$
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1answer
24 views

Projective transformation a parabola to a circle

Take the parabola $x^2 - y = 0$ in the cartesian plane. I'm not entirely sure about this, but we can express this using homogenous coordinates as $X^2 - Y = 0$ (the $W$ coefficient is $0$?) With the ...
3
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1answer
18 views

Unit velocity parametrization of a parabola.

I have a parametrization for the parabola $y = x^2$ given by: $$x(t) = t$$ $$y(t) = t^2$$ However, this doesn't have constant unit velocity, since $$\sqrt{x'(t)^2 + y'(t)^2} = \sqrt{1 + 4t^2} \neq ...
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Conicoid 3-D geometry

Please make a figure explaining question :The normal at any point P of the ellipsoid x2/a2 + y2/b2 + z2/c2=1 meets the principal planes in G1,G2,G3.Find PG1:PG2:PG3.
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1answer
50 views

Why are degenerate conics not projectively equivalent to nondegenerate conics?

This is what I understand about conics being projectively equivalent. Two conics $C1=V(F)$ and $C2=V(G)$ are projectively equivalent if there is an invertible matrix $A$ such that $F(X,Y,Z)=0$ iff ...
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0answers
13 views

10 dimensional ellipsoid covering a point

In a problem I have a 10 dimensional feature space.In that feature space I draw ellipsoids with the equation transpose(x-u)*A*(x-u)=1. u is a 10 dimensional ...
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2answers
42 views

Common tangent to a circle and ellipse

Hey guys i am noy able to solve this problem.So please do help me in solving this.The equation of common tangent to ellipse \begin{equation*} x^2 +2y^2=1 \end{equation*} and circle ...