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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12 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, $(x,y)$, ...
21
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1answer
425 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 ...
1
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1answer
47 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
23 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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0answers
15 views

How can I compute aspect ratio of ellipse? [on hold]

I have semi-axis of each ellipses. How can I compute aspect ratio of this group ellipse?
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0answers
30 views
+50

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.

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. I tried to solve it but failed.Can someone please help me to ...
2
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0answers
34 views
+50

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
32 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
22 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
29 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
24 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
34 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
21 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 ...
1
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1answer
28 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
36 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 ...
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2answers
31 views

Convert between parameteric ellipse equations

I have the parametric equation of an ellipse in this form: $$x(t)= a\cos(t)$$ $$y(t)=b\cos(t+\phi)$$ It's an ellipse centred about the origin, with a tilt angle. So three parameters. How can I ...
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0answers
16 views

Prove $2$ points are equidistant from the focus of the parabola. [closed]

The tangent at the point $P$ on the parabola $x=2at$, $y=at^2$ cuts the lines $y=-a$ and $y=a$ in the points $A$ and $B$ respectively. Prove that $A$ and $B$ are equidistant from the focus of the ...
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2answers
50 views

Finding parabola parameter given 2 points

How can I determine which is the directrix and the focus of a parabola and what is the distance between those points, only knowing that this parabola has its symmetry axis = OX and its passes through ...
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1answer
25 views

Write down the equation of the tangent to the parabola $x^2 =8y$ at the point $(4p,2p^2)$ on it. Full question in description [closed]

Write down the equation of the tangent to the parabola $x^2 =8y$ at the point $(4p,2p^2)$ on it. If the point $(3,1)$ is to lie on this tangent, find the values which $p$ may take. Hence deduce the ...
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2answers
36 views

Finding equation for hyperbola given 2 points and center [closed]

A hyperbola passes through (3,−2), (7,6) , its focal axis is on OX and its center is (0, 0). How can I write the equation for this hyperbola?
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2answers
39 views

$P$ is a variable point on the parabola $x^2 + 44x = y + 88$ and $Q$ is a point on the plane not lying on the parabola if $(PQ)^2$ is minimum, then?

Full question is : $P$ is a variable point on the parabola $x^2 + 44x = y + 88$ and $Q$ is a point on the plane not lying on the parabola if $(PQ)^2$ is minimum, then the angle between tangent at ...
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4answers
33 views

Find the equation of the tangent to the parabola $ 4x^2=y$ which is parallel to the line $4x+y-3=0$ [closed]

Find the equation of the tangent to the parabola $4x^2=y$ which is parallel to the line $4x+y-3=0$ Any help is appreciated. Thanks.
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1answer
36 views

Plotting a skew ellipse

I tried to make a geometers sketchpad toolkit for spherical geometry that is more exact (and easier to understand) than the present available one. but then I soon realised my background is not good ...
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2answers
31 views

Prove that the line $x-y-3=0$ is a tangent to the parabola $x^2=12y$ and find the coordinates of the point of contact. [closed]

Prove that the line $x-y-3=0$ is a tangent to the parabola $x^2=12y$ and find the coordinates of the point of contact. Also, how can you find the equation of the following tangents at the point $q$ ...
0
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1answer
11 views

How to parametrise a parabola with a specific domain

What would be the best method to find the parametric equations for the parabola $y = (x-2)^2$ over a given domain of $(2 ≤ t ≤ 5)$? The figure I've been given has the parabola starting from $(2,0)$ ...
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1answer
60 views

What surface can we slice to obtain a cubic curve?

We all know what a conic section is - a circle, a hyperbola, an ellipse, or a parabola. But what about the cubic curve? Does it not slice through some other 3D shape? If so, what is that called? What ...
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1answer
28 views

Get Distance Between Point and Side of Ellipse

So I have an ellipse where I know the two foci, the length and the width and all the relevant information. I then have a point somewhere in the ellipse. This point is known and an arbitrary angle ...
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0answers
25 views

Conic classification

I have a formula any and wonder what that is equation (hyperbola, point, lines, ellipse, parabola etc.) . However, I have doubts when I do the translation and rotation of coordinate systems. I know ...
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1answer
40 views

An elliptical field has the equation of its boundary $x^2+3y^2=3$ with A at an end of its major axis.A tower stands vertically at A.

An elliptical field has the equation of its boundary $x^2+3y^2=3$ with A at an end of its major axis.A tower stands vertically at A and from the points B,C on the boundary the angles of the elevation ...
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2answers
17 views

Find the coordinates of the point in which the tangent at the point 'p' on the parabola x=2at, y=at^2 intersects the x-axis.

Find the coordinates of the point in which the tangent at the point 'p' on the parabola x=2at, y=at^2 intersects the x-axis. I have the answer but do not know the process. THanks.
2
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3answers
21 views

Find the equation of the chord joining points $p$ and $q$ on the parabola $x=2t$, $y=t^2$ if $p$ and $q$ are the roots of the equation $t^2-4t+2=0$

I have the answer but do not know the process in achieving it. Find the equation of the chord joining points $p$ and $q$ on the parabola $x=2t$, $y=t^2$ if $p$ and $q$ are the roots of the equation ...
1
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1answer
35 views

Prove that the locus of the incenter of the $\Delta PSS'$ is an ellipse of eccentricity $\sqrt{\frac{2e}{1+e}}$

Let $S$ and $S'$ be the foci of an ellipse whose eccentricity is $e$.$P$ is a variable point on the ellipse.Prove that the locus of the incenter of the $\Delta PSS'$ is an ellipse of eccentricity ...
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2answers
49 views

Determine the equation of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ such that it has the least area but contains the circle $(x-1)^2+y^2=1$

Determine the equation of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ such that it has the least area but contains the circle $(x-1)^2+y^2=1$ Since the area of ellipse is $A=\pi ab\Rightarrow ...
2
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4answers
441 views

Finding Volume of Rugby Ball

I am asked to find the volume of rugby ball whose surface is given by the ellipsoid: $$\frac{x^2}{4} + \frac{y^2}{4} + \frac{z^2}{9} = 1$$ I am having trouble figuring out which coordinate system I ...
2
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1answer
50 views

A circle's sine wave is an ellipse's…

We all know what a sine wave is, and how it relates to a circle. What is the vertical and horizontal distance when I take a point and drag it along the perimeter of the ellipse? It definitely has to ...
3
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2answers
43 views

Confusion with the eccentricity of ellipse

Confusion with the eccentricity of ellipse. On wikipedia I got the following in the directrix section of ellipse. Each focus F of the ellipse is associated with a line parallel to the minor axis ...
2
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2answers
178 views

Given 3 spheres, find the equation of the plane that touches each of the spheres on the same side..?

I have a problem I am trying to solve, but I have no idea how to solve it. If I have 3 spheres, $A(1, 2, 0), B(4, 5, 0), \text{ and } C(1, 3, 2)$ of radius 1, how would I go about finding the ...
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1answer
16 views

Concyclic Eccentric angles of an ellipse.

If $\;\alpha, \; \beta,\; \gamma,\; \delta\;$ are eccentric anlges of four conclyclic points on the standard ellipse $\; \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ . Then $\alpha + \beta + \gamma + ...
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1answer
29 views

Prove synthetically: projection of a circle onto a plane is an ellipse

I am wondering how I can prove synthetically that the projection of a circle onto a plane is an ellipse.
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1answer
82 views

How to find equation of hyperbola given foci and a point?

I am currently studying multivariate calculus at university, and ive been given some practice problems before the first assignment. The problem is: A hyperbola may be defined as the set of points ...
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0answers
31 views

Solve for initial velocity - trajectory of projectile, must hit target

This has been bothering me for weeks now so I thought it was time to ask for expert help! I'm trying to develop this code for use in a game but it should still be a simple physics equation. Q: ...
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2answers
17 views

About the coordinates of an axis-aligned bounding box of an ellipse.

I have an ellipse whose center is $(c_{x},c_{y})$ and whose orientation with respect to the positive x-axis is $\theta$. Its semi-major and semi-minor axes are $a$ and $b$. My problem is that, how ...
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1answer
10 views

Finding formula for set of ellipses

I'm looking for a formula for a set of ellipses lying on the intersections of two set of circles. The python code for the two sets of circles is as follows: ...
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1answer
39 views

How to calculate a, b, center of a ellipse with given bonding box of an arc [closed]

How to calculate the ellipse a, b, center, if only the bounding box of an arc, a start point and end point given. For example: (the angles are right = 0°, left=180°, top: minus, bottom:plus) Ellipse: ...
2
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1answer
31 views

Question related to elliptical angles

Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 $ be an ellipse and $AB$ be a chord. Elliptical angle of A is $\alpha$ and elliptical angle of B is $\beta$. AB chord cuts the major axis at a point C. Distance ...
5
votes
1answer
84 views

Entire function assume real values for $z=x^2+ix$

Let $f$ be an entire function such that $f(z)$ is real for $z=x^2+ix$. Is there exists such a function which is not constant? Previously I thought that ...
1
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2answers
18 views

Values of k for which the strightline $y=kx-1$ is tangent to the parabola with the equation $y=x^{2}+3$?

How can I find the values of k for which the strightline $y=kx-1$ is tangent to the parabola with the equation $y=x^{2}+3$? I used this shortcut form $c=-am^{2}$. which gives me $k=+-2$. I think I am ...
2
votes
2answers
41 views

Differentiate $y=a(x+b)^2 -8$, then find $a$ and $b$ if the parabola:

a) passes through the origin with the gradient $16$ b) has tangent $y=2x$ at the point $P =(4,8)$ for a) $y'= 2a(x+b)$ then $16=2a(0 +b)$ I'm not sure what to do after this or if it's even right?
0
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1answer
35 views

Spivak's approach to conic sections (Michael Spivak - Calculus - p.81.)

Conic Sections img src="http://i.imgur.com/XMFtRr9.png"/> I don't understand when Spivak says (see the picture) "we can make things a lot simpler for ourselves if we rotate everything so that this ...
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2answers
46 views

Area of the ellipse $Ax^2+2Bxy+Cy^2+2Dx+2Ey+F=0$

Prove that the area of the ellipse $$Ax^2+2Bxy+Cy^2+2Dx+2Ey+F=0,$$ where $AC-B^2>0$, is equal to $$S=\frac{- \pi \Delta}{(AC-B^2)^{3/2}},$$ where $$\Delta ...