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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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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0answers
9 views

Location of foci with respect to center in a hyperbola as $a$ becomes increasingly smaller with respect to $b$

This question really confused me on the last test I took. "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 ...
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2answers
34 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 ...
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0answers
31 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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1answer
25 views

find equally spaced points on parabola

I'm trying to find equally spaced points on a parabola simply defined by $$y = \frac{x^2}{2 p}$$ Someone told me there is an easy way to split the parabola but he didn't tell me how and I cannot find ...
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0answers
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Roll one ellipse on another: Locus of center ever a circle?

Let $E_1$ be an ellipse fixed in the plane. Let $E_2$ be a second, possibly different ellipse, which rolls around without slippage outside $E_1$, touching perimeter-to-perimeter. Let $c_2(t)$ be the ...
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1answer
7 views

Constructing a conic from two point-polar pairs

Suppose I have two points $A$ and $B$ and two lines $a$ and $b$ in the (projective) plane. Can I construct a conic section for which $a$ is the polar of $A$ and $b$ is the polar of $B$? How unique ...
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1answer
25 views

Find a projective transformation that sends the locus $x^2-y^2=z^2 $ to $yz=x^2$

So I'm trying to find a function that sends $x^2-y^2=z^2$ to $yz=x^2$. I know it can be done because all conics are projectively equivalent. I think I have to use a matrix of some kind but I don't ...
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0answers
21 views

Given five points and a line find the points of the line that lie in the conic through the five points [on hold]

So I'm given 5 points in general position and a line, I already know the method using Pascal's theorem to find points in the conic but I dont know how to find specifically the ones that lie on a given ...
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1answer
34 views

Calculating semi-minor axis of an ellipse

I'm coding a solar system animation and so far it's done, but the the orbits of the planets are circular. To make the simulation more realistic, I want to use elliptic orbits. So I visited Mercury ...
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1answer
15 views

The use of the distance function for finding the end points of an ellipse

This is a reference to this question: Converting a rotated ellipse in parametric form to cartesian form. In the answer, it is posted that to find the extreme points of an ellipse, the distance ...
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3answers
27 views

Formula of parabola from two points and the $y$ coordinate of the vertex

The parabola has a vertical axis of symmetry. Given two points and the $y$ coordinate of the vertex, how to determine its formula? For example:
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0answers
9 views

Locus of mid-points of the chords of a hyperbola parallel to a certain line [closed]

Find the locus of the middle points of the chords of the hyperbola $3x^2-2y^2+4x-6y=0$ parallel to the line $y=2x$.
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2answers
20 views

Equation of normals at the end of variable chord of parabola $y^2-4y-2x=0$

Here is my problem: If the normals at the ends of a variable chord PQ of the parabola $y^2-4y-2x=0$ are perpendicular then the tangents at P and Q will intersect at?? The correct answer is $2x+5=0$. ...
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2answers
33 views

Transform of the Cartesian plane that maps hyperbolic arcs $xy = C$ to line segments

I have the finite set of curves: $$y = \frac{C}{x}, \qquad C = 2, 3, \ldots, C_{\max},$$ with $C$ and $x$ positive integers, $2 \le x \le C$ ($x$ varies on a finite domain). Is it possible to apply ...
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2
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25 views

Singular Conics and Intersection of Line with a Conic

I've been working through Silverman and Tate's book Rational Points on Elliptic Curves. They use conic equations as an introduction to singular/nonsingular curves. I've reproduced the problem with my ...
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33 views

What's an elegant expression for a general conic using complex numbers?

A 'nice' form of writing a line through $2$ points $z_1,z_2\in\mathbb{C}$ is $$z\overline{z_1-z_2}+z_1\overline{z_2-z}+z_2\overline{z-z_1}=0$$ Now I'm asked to give a similar expression for a general ...
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1answer
32 views

Why are hyperbolas defined by two branches?

Why are hyperbolas defined by two branches, unlike a parabola which only have one? Geometrically, it looks like a slice. When plotted on a graph, it's two separate curves. Why? We were never taught ...
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0answers
21 views

Finding point on ellipse given an arc length

Given a parametric representation of an ellipse: $$ x = a\cos t \\ y = b\sin t $$ Say I have a known point $P_0$ at $t = t_0$. Given also a known arc length $d$ on the ellipse: $$ d = ...
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4answers
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Find the sum of the roots of a quadratic function given the vertex of its graph

Question: At this parabola $$y = ax^2 + bx - c$$ and vertex is $T(3,9)$. What is the sum of roots of this parabola ? Help or give a hint. Thanks
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0answers
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Getting the celestial cone back from its conic section

Find the semi-vertical angle $\alpha $ of a right circular cone with z-axis symmetry cut by a plane making inclination $ \beta $ to z-axis producing the following projection on xy plane : $$ (1- ...
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2answers
39 views

Affine transformation that sends a conic to itself but does not preserve the focci or the axes [closed]

So I'm trying to find an affine transformation that sends a conic to itself but does not preserve the foci or the axes. I don't know if this can be done. I'm pretty sure that if it is possible then I ...
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1answer
40 views

Equivalence of focus-focus and focus-directix definitions of ellipse without leaving the plane

Take a look at the following two definitions of ellipse: For some fixed points $F_1,F_2$ and real number $2a>|F_1F_2|$ an ellipse is the locus of points $P$ such that $|F_1P|+|F_2P|=2a$. ...
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35 views

Conic property pedal length and polar/tangent rotations

From standard Newtonian form for focal conics $ p/r = ( 1- \epsilon \cos \theta), $ I obtained by differentiating with respect to arc: $$ \dfrac{FN}{p} = \dfrac{\cos \phi}{\sin \theta}. $$ ...
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3answers
31 views

Generating a Conic Section From 5 Points

I'm trying to generate a round trailing edge for an airfoil with either no trailing edge or a sharp trailing edge. I do this by chopping off the end of the airfoil, taking 2 points each from the upper ...
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35 views

Conic ( Parabola by looking at the equation ? )

A conic has equation given below. If the focus point is at (F, 0) then what is the value of F to 2 decimal places? $$ 10y^2-320x=0 $$ $$ ∴ 10y^2=320x $$ $$ y^2=32x ∴ y^2=4(8)x $$ $$ where, a=8 $$ ...
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1answer
24 views

Formula of finding equation of tangent line of a parabola

I have homework question. The question is The equation of tangent line of a parabola that has equation $y=Ax^{2}+Bx+C$ and parallel to $Ay=Bx+C$ line is ... I know, to solve it with using formula ...
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0answers
36 views

Volume of an ellipse rotated about a line

The question is: Find the volume enclosed by the ellipse $$9x^2+4y^2=36$$ after it has been rotated about the line $$2x+y=1$$ Basically, I don't really know where to go. I tried rotating the ellipse ...
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1answer
33 views

Finding tangent's equation that touchs parabola at $(4, 4)$

$y^2 = 4x$ is equation of a parabola. What is the equation of the tangent which touchs parabola at $(4,4)$ ? I don't know how to solve it, please help. (Excuse my bad grammer. Hope you understand ...
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0answers
28 views

Can I generate a skewed ellipse tangent to two points?

I'm trying to write a python script to generate a trailing edge (TE) for an airfoil with no TE. Basically want to make a smooth round-off nose profile to the right, the closure line should come out ...
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2answers
52 views

Calculating the length of the semi-major axis from the general equation of an ellipse

What is the most accurate way of solving the length of the semi-major axis of this ellipse? $-0.21957597384315714 x^2 -0.029724573612439117 xy -0.35183249227660496 y^2 -0.9514941664721085 x + ...
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1answer
32 views

What kind of line does this equation represent?

$x^2 – y^2 = -1$ . I know it is a hyperbola, but i want to know to reach this conclusion, (sorry for the symbols but I do not know how to use MathJax).
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1answer
26 views

need help to understand answer

Write the equation of a parabola with a vertex at $(-5, 2)$ and a directrix $y = -1$. i got $(y-2)= \frac{1}{4} (x+5)^2$ Correct answer is $(y-2) = \frac{1}{12} (x+5)^2$
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1answer
55 views

Derivation for the length of a parabola.

$$ \int_{x_1} ^{x_2}\sqrt{1+f^{'}(x)^2}dx$$ I would separately determine limits $x_1, x_2 $ as well as $x_3$(vertex) of the parabola $y= a x^2+b x+c$ getting length before inserting limits: ...
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1answer
111 views

What is the reason behind the Pythagorean relation in a hyperbola?

I am currently (in my Pre-Calculus course) deriving the equations of the conic sections. I very much understand how the relationship, in an ellipse, between $a, b$, and $c$ is established. Knowing ...
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49 views

Tangents are drawn from any point on a hyperbola, to a circle. Find the locus of mid points of the chord of contact.

Tangents are drawn from any point on the hyperbola $\dfrac{x^2}{9}-\dfrac{y^2}{4}=1$, to the circle $x^2+y^2=9$. Find the locus of mid points of the chord of contact. Attempt:- Taking the point to be ...
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1answer
34 views

Compute center, axes and rotation from equation of ellipse

Suppose I have the equation of an ellipse, in its implicit form $$Ax^2 + By^2 + Cxy + Dx + Ey + F = 0$$ For example the following: $$4.36\,x^2 + 2.89\,y^2 - 5.04\,xy + 30.8\,x - 0.6\,y + 81 = 0$$ ...
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1answer
20 views

Prove that only one normal to the parabola $y^2=4(x-11)$ passes through the focus $(12,0)$

question on the title, thanks!! I think it has to do with the normal gradient equation, which i believe is $y-y^*=-\frac y2(x-x^*)$ I have no clue what to do next. :(
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Graphing With Conics.

I have a project in math where I must create a picture using conics, with my graphing calculator. However, the equations I have found to form a picture are not in $y=\ldots$ form. How do you put the ...
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31 views

Showing a hyperbola in polar form approaches two asymptotes

Consider a curve given in polar coordinates by $r(θ)=\frac{1}{1+e\cos\theta}$, where $e≥0$. When $e>1$, show that the curve approaches two asymptotes, find them and sketch the curve. Hint: If the ...
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66 views

Polar representation of conic sections $r(\theta)=\frac1{1 + e \cos\theta}$

Consider a curve given in polar coordinates by $r(\theta) = \dfrac1{1 + e \cos\theta}$, where $e\ge0$. a) Show that the distance of each point on this curve to the line $x=\frac1e$ is a constant ...
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1answer
36 views

Reflection Of Conic Section About A Line

If a certain conic section $$ ax^2+2hxy+by^2+2gx+2fy+c=0 $$ is reflected about any line $y=mx+n$ what will be its new equation?
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1answer
24 views

A proof on the center of curves I am unsure of

Here is a proof in a book I am reading. It seems fairly short, but I kind of got lost. Especially when $\lambda$ was introduced. I usually get ideas after awhile of staring at it, but I am getting ...
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2answers
22 views

Conics - required to show $SR \times S'R' = b^2$

Consider the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b > 0$. $R$ and $R'$ are the feet of the perpendiculars from the foci $S$ and $S'$ on to the tangent at ...
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2answers
45 views

Concurrent Normals to a parabola

Let $A, B, C$ be three points on the graph of $y=x^2$, so that the normals at $A, B, C$ to the graph of $y=x^2$ are concurrent. Let $P$ be the point of concurrence. Then find the possible values of ...
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0answers
9 views

How can I interpolate between two points on an ellipse given only the two points in polar coordinates and the ratio of a and b?

If you have two points in polar coordinates, $p_1$ and $p_2$, and you have a ratio $k = a/b$ ( where a and b are parameters of an equation for an ellipse ), how can you find the radius for a point $p$ ...
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0answers
15 views

Three normals to a parabola [duplicate]

Let $A, B, C$ be three points on the graph of $y=x^2$, so that the normals at $A, B, C$ to the graph of $y=x^2$ are concurrent. Let $P$ be the point of concurrence. Then find the possible values of ...
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1answer
17 views

Polar equation for an ellipse that is not centred at the origin

Wikipedia says the polar form of an ellipse centred at the origin is What if the ellipse is not centred at the origin? Like its centred at (3, 4)?