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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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
53 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
36 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
34 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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1answer
29 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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1answer
27 views

sum of the squares of the reciprocals of the two parts of the focal chord of a parabola

Find the sum of the squares of the reciprocals of the two parts of the focal chord of a parabola. My attempt: Let $y^2=4ax$ be a parabola. Let PQ be the focal chord through the focus S$(a, 0)$ ...
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0answers
38 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
35 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
68 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
34 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
27 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: ...
8
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1answer
133 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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1answer
46 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
32 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
73 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
37 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
24 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
47 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
12 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
19 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
85 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 , ...
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1answer
25 views

Identifying elliptic curve according to equation.

To which parameter C does the following equation: $$(2-c)x^2+(3-c)y^2+2x+8y+5=0$$ is an equation for: 1) Ellipse OR Circle 2) Hyperbole 3) Parabola Well, as far as I know that in order to get ...
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1answer
61 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
43 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
21 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 ...
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2answers
68 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
126 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. ...
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1answer
24 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 ...
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0answers
47 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
24 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
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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
36 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
31 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
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1answer
43 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
54 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
23 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
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2answers
643 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 ...
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1answer
54 views

Equation general solution of intersection of two elipse

I have two elipse. E1: $\dfrac{(x-x_1)^2}{a^2}+\dfrac{(y-y_1)^2}{b^2}=1$ and E2: $\dfrac{(x-x_2)^2}{c^2}+\dfrac{(y-y_2)^2}{d^2}=1$. Please help me what is Equation general solution of ...
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2answers
123 views

Given a drawing of a parabola is there any geometric construction one can make to find its focus?

This question was inspired by another one I asked myself these days Given a drawing of an ellipse is there any geometric construction we can do to find it's foci? I think this is harder, I can't ...
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1answer
33 views

Relation between a differential equation satisfied by parabolas and a formula for the slope of their tangents

Statement 1: The slope of the tangent at any point P on a parabola, whose axis is the axis of x and vertex is at the origin, is inversely proportional to the ordinate of the point P. Statement 2: The ...
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1answer
165 views

Ellipse inscribed on a quadrilateral

The problem is: Given that an ellipse is inscribed on a convex quadrilateral and each one of it's diagonals pass through one foci of the ellipse show that the product of the opposite sides ...
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2answers
85 views

Ellipses Conics Inquiry

A carpenter wishes to construct an elliptical table top from a sheet 4ft by 8ft plywood to make a poker table for him and his budies. He will trace out the ellipse using the "thumbtack and string" ...
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1answer
27 views

Unique Specification of Ellipse Given Two Arbitrary Axes Lengths and Axes Orientation?

Without loss of generality, let an ellipse be centered on the origin (0,0) with the major axis aligned with the 45 degree line (y=x). Given the lengths of the major and minor axes, the ellipse is ...
2
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1answer
79 views

Given a drawing of an ellipse is there any geometric construction we can do to find it's foci?

For example if we're given a drawing of a circle, we can take three different points on it, draw the perpendicular bisectors of them and the intersection point is the center. Is it possible to find ...
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0answers
26 views

Making Homogenous Parabola Equation

Find the locus of the mid-points of the chords of the parabola $y^2=4ax$ which subtend a right angle at the vertex of the parabola. Now we say $y^2=\frac{4ax(yk-2ax)}{k^2-2ah}$ coefficient of ...