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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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, ...
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4answers
161 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 ...
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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 ...
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2answers
485 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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240 views

question related to ellipse

how are you?i need one help and please give me some hint to solve it,namely we should construct equation of such ellipse,focuses of which are on the axis of abscisa and are symmetrical to ...
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1answer
470 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
56 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 ...
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3answers
11k 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
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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
77 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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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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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
661 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
108 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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2answers
605 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 ...
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1answer
735 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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2answers
457 views

Find the centre, foci and asymptotes of the hyperbola

Hyperbola: $ 4x^2 - 8x - y^2 + 6y - 1 = 0 $
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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
587 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 ...
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330 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
565 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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2answers
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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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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 $A,B,C$ ...
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2answers
582 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.
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1answer
578 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 ...
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1answer
203 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
748 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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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 ...
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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
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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
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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
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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
575 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
64 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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110 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
232 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
130 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 ...
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1answer
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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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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 ...
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2answers
4k views

Parametric equation of a cone

I usually use the following parametric equation to find the surface area of a regular cone $z=\sqrt{x^2+y^2}$: $$x=r\cos\theta$$ $$y=r\sin\theta$$ $$z=r$$ And make $0\leq r \leq 2\pi$, $0 \leq \theta ...
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4answers
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Equation of a parabola

I have trouble grasping parabolas, and mainly the cartesian equations describing the,. In my mind, there are 4 possible parabolas, a parabola shaped like a mountain ($\cap$), a parabola shaped like a ...
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Foci of Ellipse - given: Width and Height

Can you help me out with the next problem. I have an ellipse based on a width and a height. Is there any way you can find out where the focal points are? I need this information because I need to ...
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Length of a Parabolic Curve

I just wanted to know how I can find the length of a curve given by $f(x) = x^2$ from $x=0$ to $x=1$. For appproximation, the length is a bit larger than the hypotenuse of isosceles right triangle ...
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Possibly flawed calculus homework question (tangent to ellipse)

I have an online homework program called Web Assign for my calculus course. It has given me this problem: Find equations of both the tangent lines to the ellipse x^2 + 9y^2 = 81 that pass through the ...
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How to get the center and the axes of an ellipse

Get the center and the semimajor/semiminor axes of the following ellipses: $$x^2-6x+4y^2=16$$ $$2x^2 - 4x+3y^2+6y=7$$ How would one get these? I have no clue. I have a problem with merely rewriting ...
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3answers
311 views

The equation of an ellipse

I have a couple of questions regarding ellipses. Get the equation of the ellips With Foci $(\pm 3,0)$ and which goes through $(2,\sqrt{2})$. This one I didn't understand AT ALL. I need some ...
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1answer
195 views

Equation of a parabola: Translations and directrixes

Find the equation of the paraboles, with: Focus $(3,0)$ and $x=-3$ is the directrix Focus $(0,2)$ and $y=-2$ is the directrix Vertex (I believe it is the vertex, the lowest/highest point) $(1,2)$ ...
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Apollonius ellipse equation $y^2=x \left( p-\frac{p}{2a}x \right) $ to standard form.

I am looking for a way to understand the last steps found at this site: http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=196&bodyId=203 The page finishes with showing in ...
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191 views

Parameters for an ellipse given measures of ellipticity

I am trying to visualize some data in the form: ...
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245 views

(Calculus 3) Having trouble finding the polar equation of a hyperbola.

Eccentricity e=sqrt(2), and one vertex is located at (2,0). I do know that if the vertex is located at (2,0), then the directrix is 2 units from the vertex. I am not sure how to find the location of ...