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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15 views

Equation of a perpendicular line through the tangent of an ellipse [on hold]

Slope of the tangent is $$ \frac{b^2 x_1}{a^2 x_0} $$ Given coordinates on the perpendicular line $= (\sqrt{a^2-b^2},0)$
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2answers
19 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
28 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
8 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
13 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
13 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
54 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
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1answer
47 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
23 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
10 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
61 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
110 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
21 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
25 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
19 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 ...
2
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2answers
27 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
33 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
23 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
17 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
48 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
19 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
557 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
52 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
104 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
21 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 ...
7
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1answer
146 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 ...
0
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2answers
26 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
22 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 ...
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1answer
67 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
22 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 ...
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2answers
139 views

Arguably the world's first differential equations

EDIT4: start of context Apologies about context. I thought that it is an all too well known reference for re-counter on the topic of differential equations. In the classical dynamic solution of ...
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1answer
11 views

Find the equation of the hyperbola that satisfies this condition

Focus is at $F\equiv(−3−3√13, 1)$, asymptotes intersect at the point with coordinates $(−3, 1)$ and one asymptote passes through $(1, 7)$ I've solved some problems that involve equations of ...
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0answers
31 views

Find the parabola given two endpoints and the midpoint along the curve

It has arbitrary orientation in 2D. I thought to equate the formulas for the arc lengths (s) between the midpoint and each end point from ...
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6answers
199 views

Area enclosed by the graph of $13x^2-20xy+52y^2+52y-10x=563$.

Find the area enclosed by the graph of $13x^2-20xy+52y^2+52y-10x=563$. First I saw that this cannot be a circle ($xy$ term), and it cannot be an ellipse with axes parallel to the coordinate axes. But ...
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2answers
54 views

Fast method to find the tangent line to a conic section: why does it work?

My teacher taught me this fast method to determine the equation of the tangent line to a conic section. In the Netherlands this is called "eerlijk delen" or literally translated into English "fair ...
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2answers
40 views

Water Density and Fluid Force (question below) [closed]

I've been trying to study the question and the answer below. Can someone tell me how to start this problem myself? I don't understand why they named one fourth of the circle equation the whole ...
3
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1answer
42 views

How homogenization of line and curve works?

I am given a curve $$C_1:2x^2 +3y^2 =5$$ and a line $$L_1: 3x-4y=5$$ and I needed to find curve joining the origin and the points of intersection of $C_1$ and $L_1$ so I was told to "homogenize" ...
2
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1answer
25 views

Parabola properties assumptions

I am trying to model projectile trajectory but I'm having some trouble. I didn't realise parabolas are this complicated... I have some assumptions that I would like to be clarified. If I specify a ...
2
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2answers
73 views

Show that $PF.PG=b^2$ in a hyperbola

If the normal at P to the hyperbola $\frac {x^2}{a^2}-\frac {y^2}{b^2}=1$ meets the transverse axis in G and the conjugate axis in G' and CF be the perpendicular to the normal from the center C then ...
3
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3answers
107 views

Property of ellipses involving normals at the endpoints of a focal chord and the midpoint of that chord

While solving a book on ellipses, I came across the following property of an ellipse which was given without proof :- If the normals be drawn at the extremities of a focal chord of an ellipse, a ...
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0answers
14 views

Measure best fitting major and minor axis length given 3 points on an ellipse

I am trying to measure the parameters of an ellipse in an image. I have the center, the rotation of the ellipse. I am trying to find the best fitting major and minor axis length based on 3 given ...
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2answers
42 views

Parabola describing projectile motion.

I am trying to create a function that will generate a parabola that describes projectile motion. Here are my inputs: The starting x-y coordinate of the throw The initial x-y velocity vector. I ...
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1answer
18 views

Test if a point is within 2 parametric “cut-off” ellipses

I have 2 parametric ellipses, both represented using the standard parametric equation of an ellipse: $$x = h + a \cos t $$ $$y = k + b \sin t $$ Lets say that the ellipses are cut-off at (see ...
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1answer
57 views

Radius vs Radius of curvature of an ellipse

I am a bit confused by the physical meaning of radius vs radius of curvature, with regards to an ellipse. For a standard ellipse: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ In this case, the $a$ ...
0
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1answer
23 views

Is the observed widest-width of an oblate sphere constant under all rotations?

This is something which I feel intuitively is true but I'm having trouble finding a way of proving it mathematically. Given an oblate sphere, or ellipsoid, with equation $$x^2+y^2+(z^2 / c^2)=1, ...
2
votes
4answers
99 views

Area of triangle bounded by line and degenerate “crossed lines” conic

The question is Show that the two lines given by $$(A^2 - 3B^2)x^2 + 8ABxy +(B^2 - 3A^2)y^2=0$$ and the line given by $$Ax+By+C=0$$ determine an equilateral triangle of area ...
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1answer
43 views

Half parabola- $y = x^2$ - does the derivative exist at x = 0?

If we took $y = x^2$ and cut it in half by letting $x\ge 0$, does the derivative still exist at $x = 0$ or is it $\text{DNE}$? I think it's still $0$ because the function and it's derivative are both ...
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0answers
15 views

Test if a point is within a double intersected ellipse

I have a case of 4 ellipses, every 2 ellipses represent a pipe (outer and inner), and a front and back (back being occluded by the front) My question is that is there an easy way to obtain whether ...
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2answers
28 views

Circle $x^2+y^2=2$ is stretched by a scale factor $2$ parallel to the $x$-axis, find the equation of Ellipse

What is the quick method or formula to finding this answer? Also the method for finding the answer when the stretch is parallel to the $y$-axis, Regards Tom
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1answer
55 views

Rotating an ellipse about a line

I'm attempting to solve the following question: What is the volume of the region formed when the ellipse $9x^2+4y^2=36$ is revolved around the line $2x+y=5$? My try: $$9x^2+4y^2=36$$ $$y ...