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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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 $|F_1P|+|F_2P|=2a$. ...
1
vote
1answer
292 views

Volume of ellipsoid bounded by two planes.

I need to find the volume of ellipsoid: $$5x^2 + {y^2\over25} + {3z^2\over4} = 1$$ if the ellipsoid is bounded by $z={-1\over2}$ and $z=1$ planes. I was able to find the total volume of the ellipsoid ...
2
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0answers
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
votes
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 ...
2
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3answers
1k views

Proving a property of an ellipse and a tangent line of the ellipse

Suppose that there is line $l$ that is tangent to an ellipse $A$ at point $\,P\,$. The ellipse has the foci $F'$ and $F$. One then creates two lines - each from each focus to the tangency point ...
34
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3answers
2k views

Fascinating Lampshade Geometry

Today, I encountered a rather fascinating problem in a waiting room: Notice how the light is being cast on the wall? There is a curve that defines the boundary between light and shadow. In my ...
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 ...
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0answers
20 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 [closed]

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 ...
1
vote
1answer
372 views

The reflective property of ellipses

I have following question to proof: An ellipse is revolved about its major axis to generate an ellipsoid. The inner surface of the ellipsoid is silvered to make a mirror. Show that a ray of light ...
0
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1answer
293 views

Determining the direct and transverse tangent lines for two non-overlapping ellipses

I am trying to determine the direct and transverse lines for two non-overlapping ellipses. I specifically mean that the two ellipses are totally separated from each other with no shared regions. I ...
9
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3answers
520 views

Equal angles formed by the tangent lines to an ellipse and the lines through the foci.

Given an ellipse with foci $F_1, F_2$ and a point $P$. Let $T_1, T_2$ the points of tangency on the ellipse determined by the tangent lines through $P$. Show that $\widehat {T_1 P F_1} = \widehat {T_2 ...
0
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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- ...
2
votes
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
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: $$ \dfrac{FN}{p} = \dfrac{\cos \phi}{\sin \theta}. $$ ...
1
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2answers
38 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 ...
1
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2answers
210 views

rotation of conic sections

In the discriminant test of conic sections(rotations), why we're checking with $B^2-4AC$. How $B^2-4AC=B'^2-4A'C'$, where $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ is changed to $A'x^2+C'y^2+D'x+E'y+F'=0$ using ...
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? $$ 10y^2-320x=0 $$ $$ ∴ 10y^2=320x $$ $$ y^2=32x ∴ y^2=4(8)x $$ $$ where, a=8 $$ ...
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3answers
163 views

Detect if two ellipses intersect

I have seen a lot of papers on how to find points of intersection between two ellipses for 2D case, but i only need to check if two ellipses are in collision. I don't need to know points of ...
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 ...
0
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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 ...
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2answers
1k views

Relationship between ellipsoid radii and eigenvalues

I should start by saying that I haven't done algebra for very long time. I recently have some work related to algebra, so I need some help to speedup. I went through a theorem in the book stating the ...
0
votes
1answer
881 views

length of the focal chord

Paragraph: $PQ$ is a focal chord of the parabola: $y^2=4ax.$ The tangents to the parabola at the points $P$ and $Q$ meet at point $R$ which lies on the line $y=2x + a.$ Question: Find the length ...
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2answers
87 views

Area inside an ellipse

Given the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, $ \ 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 ...
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vote
1answer
668 views

How to compute the chord length of 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 ...
0
votes
1answer
700 views

Calculating Intersection of an Ellipse and a Line

I found this page which gave me some equations on solving the intersection of a line with an ellipse given a point on the line and the slope of the line: There Isn't much explanation but ...
0
votes
1answer
440 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
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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 ...
0
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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).
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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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0answers
69 views

Can Fermat's descent for $x^4+y^4=z^2$ be interpreted on a conic?

Fermat proved the Diophantine equation $$(x^2)^2 + (y^2)^2 = z^2$$ has only solutions $(0,0,0)$, $(0,\pm 1,\pm 1)$ and $(\pm 1,0,\pm 1)$ using "infinite descent". The conic $C : X^2 + Y^2 - 1$ has a ...
2
votes
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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2answers
356 views

Bezier curvature

I'm trying to understand quadratic Bezier curves but I cannot get pass one thing. Please, what is a "curvature" and how can I calculate it? I'm asking because I found for instance: ...
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
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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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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 ...
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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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vote
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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2answers
188 views

A circle touches the parabola $y^2=4ax$ at P. It also passes through the focus S of the parabola and int…

Problem : A circle touches the parabola $y^2=4ax$ at P. It also passes through the focus S of the parabola and intersects its axis at Q. If angle SPQ is $\frac{\pi}{2}$ find the equation of the ...
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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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1answer
323 views

Finding the volume of a cone by integration of parabolic conic sections

I am working on a purely academic way of finding the volume of a right circular cone of height $h$ and radius $r$, (assume $h > r$), using integration of parabolic conic sections (conic sections ...
1
vote
1answer
436 views

Maximum/Minimum of Curvature - Ellipse

Find the sum of the maximum and minimum of the curvature of the ellipse: $9(x-1)^2 + y^2 = 9$. Hint( Use the parametrization $x(t) = 1 + cos(t)$) Tried to use parametrization like that, but then ...
1
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1answer
59 views

Quadric and tangents planes

Let $Q$ be the quadratic $x^2 + 4xy - 2y^2 + 6z^2 + 2y +2z = 0$ Prove that $Q$ is a cone and find its vertex. Write the tangent plane $A$ to the cone in $(0,0,0)$ and say which kind of conic is the ...