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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2
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19 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 ...
3
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0answers
25 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 ...
5
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1answer
27 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 ...
1
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0answers
19 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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0answers
10 views

Work-integration problem for a parabolic trough [on hold]

A trough is 4 feet long and 1 foot high. The vertical cross-section of the trough parallel to an end is shaped like the graph of y=x4 from x=−1 to x=1. The trough is full of water. Find the amount of ...
2
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4answers
49 views

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
19 views

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- ...
1
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2answers
37 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 ...
2
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1answer
24 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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0answers
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: $$ \dfrac{FN}{p} = \dfrac{\cos \phi}{\sin \theta}. $$ ...
2
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3answers
30 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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0answers
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? $$ 10y^2-320x=0 $$ $$ ∴ 10y^2=320x $$ $$ y^2=32x ∴ y^2=4(8)x $$ $$ where, a=8 $$ ...
0
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1answer
23 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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
2answers
49 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 + ...
1
vote
1answer
31 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).
0
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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 $(y-2) = \frac{1}{12} (x+5)^2$
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: ...
8
votes
1answer
103 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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0answers
45 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
26 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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0answers
12 views

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 ...
0
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0answers
27 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 ...
4
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0answers
65 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
21 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
43 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 ...
0
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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
14 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 ...
1
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1answer
16 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)?
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2answers
64 views

Converting a rotated ellipse in parametric form to cartesian form

I have a rotated ellipse in parametric form: $$\begin{pmatrix}y \\ z\end{pmatrix} = \begin{pmatrix}a\cos t + b\sin t \\ c\cos t + d\sin t\end{pmatrix} \tag{1} $$ or, $$(y,z) = (a\cos t + b\sin t , ...
2
votes
1answer
54 views

Geometric proof of this property of the ellipse

I came across the following property of the ellipse: The distance from a focus of an ellipse to any point on the ellipse is equal to $a(1-e \cos\theta)$. Where the $a$ is the length of ...
2
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1answer
34 views

ratio of semi-minor axis to semi-major axis

So I'm writing a paper for a math class on Kepler's equation, and I've ran into a snag on deriving the equation. I've been mostly following the book Solving Kepler's Equations by Peter Colwell. I ...
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0answers
14 views

Find the distance between two points on a curve (between two IMU sensors)

I have an elastic belt with six sensors on it. Each sensor contains a gyroscope and an accelerometer. I know the problem of finding the distance between two points on a curved surface has been asked ...
4
votes
2answers
67 views

Finding axis of ellipse described by $x=a\cos t+ h\sin t$,$ y=b\sin t + g\cos t$

Hi I am in need of help here for my project. Basically I have managed to obtain this form of equation. Example: $a=-181,h=33,b=185.9$ and $g=18.3$. When I plot it on a graphing program, it looks like ...
2
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2answers
119 views

Finding an ellipse knowing two points and the arc length

I have two (Cartesian) points of an elipse, and I know the arc length between them, but I don't know either radii or where the centre is. I know that one known point lies on the minor radius though. ...
0
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1answer
22 views

Ellipses Conics Proof

We are covering conics in our school and we just finished the ellipse section. An ellipse, by definition, is the "set of points such that the sum of the distances from any point on the ellipse to two ...
0
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0answers
37 views

Focus And Vertex Of An Inclined Parabola

How to find focus,vertex,directrix of a parabola like $x^2+y^2+2xy-6x-2y+3=0$. Well i know how to find those for any parabola of form $y^2=4ax$ but im just not being able to figure out a way to ...
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1answer
20 views

Determine the equation of a hyperbola with foci at $(3,7)$ and $(3,−1)$ and with eccentricity $e=2$.

Determine the equation of a hyperbola with foci at $(3,7)$ and $(3,−1)$ and with eccentricity $e=2$. If someone could check my answer that would be great. By looking at the foci it is easy to deduce ...
4
votes
2answers
32 views

How to identify properties of conic $12x+y^2-6y+45=0$

I need to find out the type of conic, the coordinates of the center, focus (foci), vertex (vertices), directrix for the conic given by: $$12x+y^2-6y+45=0$$ I completed the square to get ...
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0answers
35 views

Residuals of ellipse fit

I'm working on a software algorithm that fits an ellipse to a number of $(x,y)$ points using the formula $$ax^2 + bxy + cy^2 + dx + ey + f = 0$$ although $b = 0$ since the ellipse is never rotated. ...
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0answers
27 views

Conormal Points Parabola

Let the line $lx+my=1$ cut the parabola at $y^2=4ax$ in the points A and B.Normals at A and B meet at a point C. Normal from C other than these two meet at D.Then coordinates of D are? I tried to ...
2
votes
1answer
30 views

Cannonball Parabola Conics Problem

I found this problem in a math textbook and I was a little confused on how to solve it. Here is the problem: A cannon fires a cannonball. The path of the cannonball is a parabola with vertex at the ...
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1answer
53 views

Is it possible to find the equation of parabola with these givens?

If I have a parabola as seen below, and I know Vmax, Vi, and the area, 'd' under the curve from x = n to x = t, is it possible to find the equation of the parabola? Or do I need more information? n ...
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1answer
21 views

Equation of an ellipse after reflection

Give the equation of the ellipse $x^2+2y^2-6x+16y+9=0$ after reflection in the line $y=-x$. I completed the square and obtained $$\frac{(x-3)^2}{32}+\frac{(y+4)^2}{16}=1$$ Now I changed $y$ and ...
5
votes
2answers
597 views

Creating a Hyperbola with a Flashlight

I ran into this problem in a textbook and was intrigued by it. Conics are generally formed through different cuts one can make with the shape of a cone. But, there have been recent discussions on ...