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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3answers
80 views

Find at least two ways to find $a, b$ and $c$ in the parabola equation

I've been fighting with this problem for some hours now, and i decided to ask the clever people on this website. The parabola with the equation $y=ax^2+bx+c$ goes through the points $P, Q$ and $R$. ...
5
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0answers
745 views

Decomposition of a degenerate conic

As it has been done for the Intersection of conics using matrix representation the aim of this page is providing an exaustive and clear numerical example that describe the math behind the ...
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3answers
560 views

Equation of one branch of a hyperbola in general position

Given a generic expression of a conic: $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F=0$$ is there a way to write an expression for one of the branches as a function of the coefficients? I tried using the ...
9
votes
3answers
583 views

Equal angles formed by the tangent lines to an ellipse and the lines through the foci.

Given an ellipse with foci $F_1, F_2$ and a point $P$. Let $T_1, T_2$ the points of tangency on the ellipse determined by the tangent lines through $P$. Show that $\widehat {T_1 P F_1} = \widehat {T_2 ...
2
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1answer
2k views

ellipse equation from eigenvectors and eigenvalues

I have a eigenvectors d1,d2 and eigenvalues v1,v2. The eigenvectors are axes of an ellipse that surrounds data points, with center u,v, and radii size of the eigenvalues. How can I find the ellipse ...
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0answers
135 views

Tangents to an ellipse

I was reading a section on conic sections in a book, and the author writes proofs that show that tangent lines to each of the three non-degenerate types of conic sections intersect at only one point. ...
8
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1answer
3k views

Intersection of conics using matrix representation

I came across a very interesting section of a wikipedia article on conics: http://en.wikipedia.org/wiki/Conic_section#Intersecting_two_conics I am trying to work out a couple of examples to add to ...
8
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1answer
24k views

Ellipse in polar coordinates

I think Wikipedia's polar coordinate elliptical equation isn't correct. Here is my explanation: Imagine constants $a$ and $b$ in this format - Where $2a$ is the total height of the ellipse and $2b$ ...
2
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1answer
177 views

Help With Plugging in Values Distance Point to Ellipse

Can someone help me with plugging in the correct values in the equations given in this thread (accepted answer) -> Calculating Distance of a Point from an Ellipse Border The result values for x and ...
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1answer
60 views

Parameters of an elliptic equation?

I know that an ellipse equation described by: $\frac {x^2} {a^2} + \frac {y^2} {b^2} =1$ My question is in the equation above how many parameters we need to estimate? Two or four? The unknows ...
2
votes
1answer
243 views

How to find the equation of circle whose diameter is the latus rectum of the parabola.

The only hint given in this question is $x^2 = -36 y$ I am having problems starting the question I am clueless how to solve it.
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5answers
182 views

If $a,b \in \mathbb R$ satisfy $a^2+2ab+2b^2=7,$ then find the largest possible value of $|a-b|$

I came across the following problem that says: If $a,b \in \mathbb R$ satisfy $a^2+2ab+2b^2=7,$ then the largest possible value of $|a-b|$ is which of the following? $(1)\sqrt 7$, ...
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1answer
78 views

Perimeter of triangle inside

Given an ellipse centered at $(3,-3)$, and has a focus at $(3,-8)$. What is the perimeter of a triangle that entirely lies within the ellipse and has two of its vertices on the foci of the ellipse ...
0
votes
1answer
429 views

In an equation that looks like the standard form of an ellipse, what must the constant on the RHS equal for exactly one solution?

I am working on a homework question: What must be the value(s) of $c$ for the following equation to have exactly 1 solution? The equation is of the standard form of the equation for an ellipse, ...
7
votes
4answers
162 views

closest point to on $y=1/x$ to a given point

I feel like I'm missing something basic - given a point $(a,b)$ how do I find the closest point to it on the curve $y=1/x$? I tried the direct approach of pluggin in $y=1/x$ into the distance formula ...
0
votes
2answers
119 views

Conics generalized to surfaces of constant curvature

Do conic sections have an interesting generalization to surfaces of constant curvature? Consider a sphere (constant positive curvature) $\mathcal{S}$ centered at $O$, as well as points $A, B \in ...
2
votes
2answers
576 views

Hyperbola is a pair of straight lines?

I'm confused by this question: If $f(x) = 2x^2 - 6y^2+xy+2x-17y-12=0$ is to represent a pair of straight lines, one of which has equation $x+2y+3=0$, what must be the equation of the other ...
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1answer
522 views

Solving a Conic Matrix given these Equations

Given a conic $\Gamma$ that has the equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, $\Gamma$ can be represented by the symmetric matrix $$\mathbf{C} = \begin{bmatrix} A & B/2 & D/2\\ B/2 & ...
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0answers
57 views

distance to sheared right circular frustum

How do I calculate the distance to a sheared right-circular frustum? In particular, I'm shearing in a direction perpendicular to the axis, so the cross sections remain parallel circles. I know I can ...
6
votes
3answers
13k views

Finding the angle of rotation of an ellipse from its general equation and the other way around

The general equation for an ellipse is $Ax^2+Bxy+Cy^2+D=0$. How do I find the angle of rotation, the dimensions, and the coordinates of the center of the ellipse from the general equation and vice ...
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4answers
1k views

Getting the equation of an ellipse using the constant and the foci

Find the equation of the ellipse with the foci at (0,3) and (0, -3) for which the constant referred to in the definition is $6\sqrt{3}$ So I'm quite confused with this one, I know the answer is ...
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0answers
78 views

The sides of a triangle touch $y^2=4ax $ and two of its angular points lie on $ y^2=4b(x+c) $

The sides of a triangle touch $y^2=4ax$ and two of its angular points lie on $y^2=4b(x+c)$. What is the locus of the third angular point?
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2answers
89 views

Co-Ordinate Geometry

P is a point which moves in the x-y plane, such that the point P is nearer to the centre of a square than any of the sides. The 4 vertices of square are (+/-a,+/-a). The region in which P will move is ...
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0answers
193 views

My solution is right and the book is wrong (parabolas) or did I misunderstand it?

Find the equation of the parabola with the vertex at the origin; directrix 2x = 3 So what I did is, find the equation of the directrix $$x = \frac{3}{2}$$ and then because its the directrix, the ...
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2answers
727 views

Formula and foci of ellipse formed by intersection of ellipsoid and plane

I have the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ and the plane $n_xx+n_yy+n_zz=0$. They intersect along an ellipse. 1) What is the formula of the ellipse, and 2) What is the ...
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1answer
115 views

Recovering a conic from a pole-polar pair

Consider a conic section $C$ in $\mathbb{R}^2$. Every point $P$ in the plane has a "dual" (pole-polar duality) line $L$ with respect to $C$ such that lines $PA$ and $PB$ are tangent to $C$, where $L ...
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votes
2answers
640 views

Find the standard form of the conic section $x^2-3x+4xy+y^2+21y-15=0$

Find the standard form of the conic section $x^2-3x+4xy+y^2+21y-15=0$. I understand the approach in trying to solve these problems. But the $4xy$ is confusing me. I am not sure of where to start on ...
1
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1answer
811 views

find a circle tangent to an ellipse

As shown in the figure, the circle is moving upwards along the line $x=x_0$ http://i.imgur.com/bEntX.png suppose we know the following parameters: $a,b,x_0,r$ The ellipse equation is ...
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1answer
1k views

Find the equation of the parabola with focus (6,0) and directrix x=0

Find the equation of the parabola with focus $\ (6,0) $ and directrix $\ x=0 $ What I have done so far: $ (x-h)^2 = 4p(y-k) $ $ (h,k) = (3,0) $ $ (x-3)^2 = 12y $ as p = 3 However, the answer ...
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1answer
603 views

how to calculate the double integral over the intersection of an ellipse and a circle

How to calculate the double integral of $f(x,y)$ within the intersected area? $$f(x,y)=a_0+a_1y+a_2x+a_3xy$$ $a_0$, $a_1$, $a_2$, and $a_3$ are constants. The area is the intersection of an ellipse ...
0
votes
1answer
446 views

Calculating the distance from a certain place to the equator

So, let's say we have a certain place on earth and we want to roughly calculate the shortest distance from that place to the equator. Is my method correct: Since the earth is roughly a sphere, we ...
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4answers
593 views

Finding Eccentricity from the rotating ellipse formula

I see that from a normal ellipse formula, we can acquire the eccentricity via this formula here. However, for this formula (1): $A(x − h)^2 + B(x − h)(y − k) + C(y − k)^2 = 1$ When parameter $B ...
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votes
2answers
2k views

The fastest way to obtain orientation θ from this ellipse formula?

In this rotating ellipse formula: $A(x − h)^2 + B(x − h)(y − k) + C(y − k)^2 = 1$ Suppose I have $A,B,C,h,k$ parameters, and I want to obtain the angle $θ$ from the centroid $(h,k)$ to the horizontal ...
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3answers
317 views

What are the A,B,C parameters of this ellipse formula?

I am looking at $$A(x − h)^2 + B(x − h)(y − k) + C(y − k)^2 = 1$$ This is a rotating ellipse formula, where $h,k$ are the centroid of the ellipse. I have tried looking around for $A,B,C$ ...
6
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2answers
4k views

A construction using straightedge and compass

Given a circle, it's easy to contruct its center. The question is: given an ellipse, draw the foci. I don't know whether it's possible to do this using only straightedge and compass.
2
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1answer
648 views

Convert linear to angular speed while moving in an ellipse

I have an ellipse, and I know everything about it (foci position, center position, a-axis, b-axis). In it, a particle is moving. I have it's angle in relation to one of the focus of the ellipse. And ...
2
votes
1answer
207 views

Help in finding curve equation.

What I have is length of the bottom line $L$ and area under parabolic curve $S$. How can I find this parabolic curve equation, depending on area under it? The following picture illustrates the ...
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2answers
821 views

Find an equation of the parabola satisfying the following properties.

latus rectum is the line segment joining the points $(2,4)$ and $(6,4)$; passing through the point $(8,1)$. Let $P$ be the point $(2,4)$ and $Q(6,4)$. The distance between the $P(2,4)$ and $(6,4)$ ...
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3answers
30 views

Question regarding tangents?

Q : Find the equation of the line that passes through the origin of the coordinations and the focus of the parabola $y=x^2+4x+1$.. so I found the focus $( -2;-3)$ and the line that passes through ...
6
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4answers
1k views

Area of an ellipse

An ellipse has equation : $$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F =0$$ Can you provide an optimum method to find it's area?
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2answers
83 views

Geometry Parabola $2x^2+\alpha x+3\alpha$ to find common point

Can you help me find the answer to this question? For any real number $\alpha$, the parabola $f_{\alpha}(x) = 2x^2 + \alpha x + 3\alpha$ passes through the common point $(a, b)$. What is the value ...
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2answers
4k views

Finding a general equation for a quadratic curve passing through three points.

I have three points (250, 0), (500,500) and (750, 0). To find a curve passing through these points all I have to do is plug-in these values into the general quadratic equation: f(x) = ax^2 + bx + c ...
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1answer
209 views

Cutting a parabola

What would be obtained on cutting the solid parabola. I searched various sites,most of them say cone. But I am unable to visualize it. Can Someone please help.
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1answer
615 views

Calculate ellipse diameters with five points (center point and four other).

Is it possible to calculate radii (or diameters) of ellipse given: ...
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0answers
66 views

Modify ellipse equation

How can one modify an ellipse equation with a Gaussian function to get a new ellipse with bump and/or valley (example)?
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2answers
116 views

Finding An Equation For A Parabola

The information given in this particular problem: Axis is parallel to y-axis; graph passes through and $(4,11)$.$(3, 4)$ $(0,3)$ From this information, I know that it opens either upwards or ...
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2answers
242 views

Derivation Of A General Equation Of An Ellipse

I am currently reading the topic alluded to in the title of this thread. In my textbook, after the equation has been derived, $\Large\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, it says by finding the ...
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1answer
131 views

Problem with ellipse equation

How one get from this ellipse equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ that ellipse equation $$\frac{x^2+y^2}{F(\phi)^2}=1,$$ where $$F(\phi)=\frac{ab}{\sqrt{(b\cos\phi)^2+(a\sin\phi)^2}}$$ and ...
0
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1answer
216 views

Is the set of points of equal distance to the surface of an ellipsoid again an ellipsoid?

Consider the hyperellipsoid $A$ in $\mathbb{R}^d$ given by the semi-major axes $a_1,a_2,\ldots,a_d$. Do points on the surface of the hyperellipsoid $A'$ with semi-major axes $a_1-\varepsilon, ...
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2answers
144 views

Can an ellipse with fixed semi-axis have different values of eccentricity?

Warning: this is probably a ridiculous question but here goes... Can an ellipse with a semi-major axis $a$ take on different values of eccentricity $e$? I have seen various places where it seems to ...