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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How to find the height ($z$) on an elliptic cone at a point $(x, y)$

I am attempting to write a java method which returns the height of an elliptic cone given a $(x, y)$ point within the base. I have an elliptic cone centred at $(x_1, y_1)$, the major axis a, minor ...
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0answers
72 views

Finding a positive definite matrix to satisfy the general equation of an ellipse

I am trying to find a matrix A such that $(1)$ can be written as $v^TAv=1$ where $v=(x, y)^T$. $(1)$: $$\left(\frac{x}{a_1}\right)^2 + \left(\frac{y}{a_2}\right)^2 - ...
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Problem in conics question

A vertical line passing through the point ($h$,0) intersects the ellipse $$\frac{x^2}{4}+\frac{y^2}{3}=1$$ at the points P & Q.Let the tangents to ellipse at P & Q meet at the point R.If ...
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2answers
65 views

Area inside an ellipse

Given the ellipse x^2/25\ + y^2/16 = 1, A = (5,0), B = (0,4); Find point C (with both coordinates positive) on the ellipse, such that the area between AC and the ellipse (S1) will be equal to the area ...
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1answer
61 views

Is this a correct way to derive the equation of an ellipse/hyperbola?

I was just testing to see if I could derive the equation of an ellipse (and consequently a hyperbola) with the least amount of information to remember. The small amount of information I chose to use ...
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1answer
92 views

Ellipse Tangents in 3D

I know that we can find the tangent of the ellipse in 2D by taking the derivative of the equation defining the ellipse. But I'm little bit confused about finding the ellipse tangent in 3D. Where the ...
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1answer
59 views

Homography between ellipses

This is a spin-off from a comment on Stack Overflow. How can I find a homography between two ellipses in the plane?
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26 views

How are the sine functions along with the hyperbolic functions visualized with imaginary rotations?

Since we know that: cos(t)=cosh(it) and isin(t)=sinh(it) I've been thinking about this, and obviously this is referring to how if you move at a right angle from a circle on a conic section, you end ...
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149 views

Area under parabola using geometry

We have to find the area of the pink region. As we all know this can be evaluated using limiting its Riemann sum, of which its a standard example. However I want to know if this can be done without ...
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1answer
38 views

to divide quarter of an ellipse into two equal halves

I wanted to divide the quarter of an ellipse into two equal halves. In what angle should I divide it so that both the arcs formed are equal in length. Finally I wanted to find the midpoint of the arc ...
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1answer
49 views

How to find position of object on a parabola?

I'm making a computer game, where an object flies along a parabola curve. This object is 'thrown' by a 'robot' towards another robot. I know the vertex of the parabola, and also it's two ...
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56 views

Finding vertex of a parabola -conflicting answers

For a given problem $g(x) = 5x^2 - 2x +1$, we were expected to describe the graph. WolframAlpha and I are finding conflicting issues. I do not think I made an approximation anywhere that would explain ...
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History behind the choice of letters $h$ and $k$ for the vertex of a parabola?

After failing to find a historical explanation for usage of letters $h$ and $k$ for the vertex of a parabola in most relatively recent textbooks in anglosphere, I turn to math.SE. Is there any ...
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1answer
66 views

Concurrency-Three parabolas sharing common directrix.

I have found this result by exploring for new problems. If three parabolas share a common directrix and each pair intersect each other in two points, then, the lines joining the two intersection ...
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2answers
58 views

Find the expression of a parabola. [closed]

I need to figure out the expression of this parabola with the points $(-2,2)$, $(0,1)$, and $(1,-2.5)$.
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3answers
89 views

General form of a circle

My math teacher taught me that the general form (equation) of a circle is: $$ ax^2+by^2+cx+dy+e=0 $$ He also asked us this: If the product of $c$ and $d$ is negative, then what 2 quadrants can the ...
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5answers
102 views

The equation of parabolas.

I have trouble grasping some basic things about parabolas. (This should be easily found on Google, but for some reason I couldn't find an answer that helped me). I know one simple standard equation ...
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1answer
47 views

Calculating a position of an object on a parabola.

I am working on a simple 2D computer game. In the game, I have a 'robot' that throws a ball towards another robot, in the shape of a parabola. Both 'robots' are positioned on the x axis, aka their y ...
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1answer
44 views

Determining center of Ellipse with limited Data Points

The dataset I am using only has 200 degrees of the ellipse. The ellipse is not centered at (0,0). The data in this case ranges ...
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1answer
128 views

Calculate Ellipse From Points?

How can I calculate an ellipse from a group of points ? Result: center point, x-radius, y-radius ? I'm not mathematician so I don't really know the best parameter style for ellipses. This ellipse ...
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1answer
69 views

Intersection of conics

By conic we understand a conic on the projective plane $\mathbb{P}_2=\mathbb{P}(V)$, where $V$ is $3$-dimensional. I'd like to ask how to find the number of points in the intersection of two given ...
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42 views

Find the eccentricity of the conic $4(2y-x-3)^2 -9(2x+y-1)^2=80$

Find the eccentricity of the conic $4(2y-x-3)^2 -9(2x+y-1)^2=80$ Solution : $4(2y-x-3)^2 = 4x^2-16xy+24x+16y^2-48y+36$ and $9(2x+y-1)^2 = 36x^2+36xy-36x+9y^2-18y+9$ $\therefore 4(2y-x-3)^2 ...
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4answers
153 views

Find $z$ such that $|z+1|+ |z-1|=4$

I have this problem: Find all points of the complex plane wich satisfy: $$|z+1| + |z-1| = 4 $$ I know this is an ellipse with foci 1 and -1, and i know the answer is : $$3 x^2+4 y^2 \leq 12$$ but ...
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1answer
63 views

Ellipse and circle

if $\alpha$, $\beta$, $\gamma$, $\delta$ be the eccentric angles of four points of intersection of the ellipse and any circle,prove that $\alpha+\delta+\beta+\gamma$ is an even multiple of $\pi$ ...
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87 views

Proving properties of an ellipse

I'm studying about ellipse and its properties. My reference is the following pdf: http://nebula.deanza.edu/~bloom/math43/ellipse-derivation.pdf My questions are from the very first page of the ...
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2answers
59 views

How to get radius at any specific point in ellipse

How to find radius of ellipse at any point $(x_1,y_1)$. We know semi-major axis and semi-minor axis i.e. $a$ & $b$. center of ellipse $(x_0,y_0)$. Somewhere I found. $$ r = \frac{ab}{\sqrt{ ...
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1answer
88 views

How to show that this line touches the hyperbola?

The question is: $PQ$ is a chord joining the points $\phi_1$ and $\phi_2$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If $\phi_1\,+\,\phi_2 = 2\alpha$, where $\alpha$ is constant, prove ...
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67 views

finding eccentricity of ellipse??

If the tangent at any point of the ellipse make an angle α with major axis and an angle β with focal radius of the point of contact then show that the eccentricity of the ellipse is given by ...
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4answers
218 views

Finding the maximum value of a function on an ellipse

Let $x$ and $y$ be real numbers such that $x^2 + 9 y^2-4 x+6 y+4=0$. Find the maximum value of $\displaystyle \frac{4x-9y}{2}$. My solution: the given function represents an ellipse. Rewriting it, we ...
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2answers
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Recognize conics from the standard equation

Suppose $Ax^2+Bxy+Cy^2+Dx+Ey+k=0$ is a conic in the Euclidean plane. How do I recognize what is it? In my book they have proved the determinant test that if $B^2-4AC$ is $>0$ if hyperbola, $=0$ if ...
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441 views

Determining the major/minor axes of an ellipse from general form

I'm implementing a system that uses a least squares algorithm to fit an ellipse to a set of data points. I've successfully managed to obtain approximate locations for the centre of the ellipse but I ...
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2answers
270 views

Length of chord on ellipse

Suppose I have an ellipse centered at the origin, preferably expressed in its matrix form, and I want to know the chord length of a segment that passes through the origin with the endpoints at the ...
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2answers
492 views

Find angle at given points in Ellipse

I have Ellipse's center-points, minor-radius and major-radius. I can find, how to check if given point(x, y) exists in Ellipse or not. Now, I want to find given point(x,y) exists at which angle in ...
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Foci Concentric Circles

My approach: Using the foci formula $$c=\sqrt{a^2-b^2}$$. By plugging in $a=3$ and $b=2$ I obtain plus and minus $\sqrt{5}$. But there's 2 choices with a root 5 result. How do i know which one is ...
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1answer
29 views

Trying to solve conic for ellipse equation

I'm trying to find out what conic the following equation represents. $9x^2+4y^2+18x-16y+24 = 0$ I know that the general ellipse equation is $(x^2)/a + (y^2)/b = 1.$ I got $9(x+1)^2 + 4(y-2)^2 = 1$, ...
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2answers
273 views

Ellipse problem : Find the slope of a common tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and a concentric circle of radius r.

Problem : Find the slope of a common tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and a concentric circle of radius r. Few concepts about Ellipse : Equation of Tangent to ellipse ...
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1answer
71 views

Coordinate System Rotation and Cross Term

If I have a conic equation $$ 5x^2 - 4xy + 8y^2 = 36 $$ and $ \left[\begin{array}{cc} 5 & -2\\ -2 & 8 \end{array}\right] $ in matrix form, whose eigenvalues are 4 and 9, how would I rotate ...
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1answer
107 views

Reflection inside an ellipse

From a typical point $P$ inside an ellipse, how many points $Q_i$ on the ellipse have $PQ_i$ normal to the ellipse? Someone asked me at school many years ago but I don't think I worked it out.
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Area of ellipse

The question is: If A represents the area of the ellipse $\,3x^2+4xy+3y^2=1$, then the value of $\frac{3\sqrt5}{\pi}A$ is For this I used rotation of axes for eliminating the $xy$ term from ...
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1answer
64 views

Conic matrix and diagonalization

If I have the conic $C$: $$ 5x^2 - 4xy + 8y^2 = 36 $$ How would I express it as a matrix of the form: $$ \begin{bmatrix}x&y\end{bmatrix} \begin{bmatrix}a&b/2\\b/2&c\end{bmatrix} ...
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2answers
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Is there a latus other than the one in the rectum?

The name "Latus Rectum" sounds so very specific. Infact when I once asked why it is called as such, an explanation stated that the concave side of a parabola is called a rectum and that latus was ...
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An Easier way to solve simple equations of this type

Im currently working with ellipses and I've been given two points on a ellipse whose major axis is along the x-axis, $(4,3)$ and $(-1,4)$. The question asks me to find the length of the major and ...
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3answers
36 views

Ellipse representation

The equation $\frac{x^2}{2-a}+\frac{y^2}{a-5} +1 = 0$ represents an ellipse if $a\; \epsilon$ (A) $(2,\frac{3}{2})\;\cup\;(\frac{3}{2},5)$ (B) $(2,\frac{3}{2})$ (C) $(1,\frac{3}{2})$ (D) ...
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1answer
53 views

How large can a circle's radius be in an ellipse?

I have an ellipse centered on the origin parameterized by $a$ and $b$. Given its $x$ coordinate, how large can its radius be and still have the circle inside the ellipse?
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28 views

ellipse chord length along its axis.

how to determine the position in an ellipse, where the chord length is equal to its minor axis and perpendicular to the major axis? Is there any equation for it?
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89 views

If the segment intercepted by the parabola $y^2 =4ax$ with the line lx +my +n=0 subtends a right angle at the vertex, then

Problem : If the segment intercepted by the parabola $y^2 =4ax$ with the line lx +my +n=0 subtends a right angle at the vertex, then (a) 4al +n=0 (b)4am +n=0 (c) al +n=0 (d) 4al +4am +n=0 ...
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161 views

The vertex of the parabola is the point (a,b) and latus rectum is of length l. If the axis of the parabola is along the positive direction of y-axis

Problem : The vertex of the parabola is the point (a,b) and latus rectum is of length l. If the axis of the parabola is along the positive direction of y-axis, then its equation is (a) $(x+a)^2= ...
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130 views

Focus-Focus Definition for a Parabola

I am looking at the focus-focus definitions of the conics, i.e. defining them as the locus of points with the property that some function of the distance from the point to two foci is a constant. ...
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1answer
55 views

Ellipse, hyperbola and principle axis

Would anyone mind telling me how to solve (a)? I have no idea what I should do to solve this problem. Also, what is principal axes?
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224 views

The chord length along 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 ...