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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1answer
46 views
Trajectory of a projectile.
From the definition of a parabola can we prove that the trajectory of a projectile is parabolic? And can this be proved by calculus?
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1answer
71 views
Finding the major and minor axis vertices for an ellipse given two conjugate diameters?
I've been googling, searching forums and looking in my old algebra/trig books to try to understand how to find the end points to the major and minor axis of an ellipse given the end points of two ...
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0answers
27 views
egg curve estimation
Let $p_{1...3}$ be three points on an ellipse, and $t_{1...3}$ be their tangent lines. For $i={1..2}$, let $M_i$ be the point of intersection of $t_i$ and $t_{(i+1)\%2}$, and $K_i$ be the midpoint of ...
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1answer
13 views
length of the focal chord
Paragraph:
PQ is a focal chord of the
parabola: y2=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 of the chord PQ.
5
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0answers
58 views
To construct an ellipse, being a projection of a great circle, given two points on it
I'm looking for a geometric construction which would allow me to draw an ellipse, which is supposed to be an orthographic projection of a great circle of a sphere, given two points on it.
The ...
1
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0answers
10 views
How can i evaluate a parabolic region in 2-Dimensions onto n-Dimensions?
I am working on the K-Nearest Neighbor algorithm and have the solution to a problem as a parabola in 2-dimensional space. Should I continue this solution onwards to n-dimensions (and thereby create ...
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vote
2answers
24 views
Minimizing area of a triangle with two fixed point and a point on parabola
A triangle is made up of three points, $A, B$, and $P$.
$A(-1, 0)$
$B(0, 1)$
$P$ is a point on $y^2 = x$
Minimize the area of Triangle $ABP$.
My approach is far too complicated, which ...
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1answer
32 views
Finding equation of parabola
I have a group of points from a graph.
When I connect the points I get a shape which looks like the one's of the function f(x) = a / x .
How can i precisely find the equation of the shape ?
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0answers
47 views
Sample Code to Generate Points on the Rim of a Randomly Rotated Cone : What's Going On Here?
Related to this question:
http://math.stackexchange.com/questions/407897/randomly-generate-point-on-shell-from-3-points-2-angles-with-uniform-angle-dis
I'm trying to reverse engineer the ...
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vote
2answers
34 views
Finding the point on a rotated ellipse corresponding to a given tangential angle
I was going to initially ask for the solution to this problem here, but I have come upon a solution by some hand derivation and wanted to verify it here. Please note that after high-school I have had ...
14
votes
4answers
162 views
A geometric reason why the square of the focal length of a hyperbola is equal to the sum of the squares of the axes.
This may be a phenomenally stupid question, so apologies in advance. But when I teach conics, I show why $c^2=a^2-b^2$ for ellipses geometrically, just by drawing the obvious isosceles triangle from ...
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0answers
214 views
How many points of intersection between an ellipse and an $L_p$-circle?
Consider an ellipse $E$ in the plane, centered at the origin. (In my case, the minor axis points into the nonnegative quadrant.)
Let S be an "$L_p$-circle": $S = \{(x,y) : |x|^p + |y|^p = 1\}$, ...
3
votes
1answer
59 views
shadow cast by a circle
A point source emits light at a circular disc (thickness negligible), and a shadow is left on a wall (XY plane) behind and parallel to the disc. The Z component of distance between the point source ...
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vote
2answers
58 views
Can I find the equation of an ellipse with these points?
How can I solve an ellipse with its major axis on the $x$-axis, given one focus, and two points on the ellipse, one of which I know to be on the major axis?
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2answers
53 views
hyperbola curve formula in 3 dimensions
Cartesian formula for 2d hyperbola curve is $x^2/a^2-y^2/b^2 = 1$.
What is the formula for a 3d hyperbola curve?
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votes
2answers
45 views
Is it possible to find out $x^2$ parabola and function from 3 given points?
I am programming a ball falling down from a cliff and bouncing back. The physics can be ignored and I want to use a simple $y = ax^2$ parabola to draw the falling ball.
I have given two points, the ...
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0answers
30 views
What is the answer? [closed]
A line is drawn through point P(-1,2) meets the hyperbola xy=c2 at the points A & B ( points A & B lie on the same side of P ) & Q is a point on the line segment AB.
1) If the point ...
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2answers
38 views
Given an semi-ellipse inscribed about a square, how do I find the equation of the ellipse?
Given the following diagram:
Where:
W = (-1, 0)
X = (-1, 2)
Y = (1, 2)
Z = (1, 0)
How can I find M?
The ellipse can be assumed to be a semi-ellipse with one of the foci on $\bar{XY}$. I'm ...
0
votes
2answers
124 views
How many times can quadric kiss cosine at given point?
Let a quadric $ax^2+2bxy+cy^2+dx+ey+f=0$ touches the plot of $y=\cos(x)$ at the point $(0,1)$ with multiplicity $n$. What is the maximum possible value of $n$? Recall that a joint point $P$ of ...
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1answer
25 views
Maximum Y in a rotate ellipse with a, b and phi
We have major axis, minor axis and the phi between major axis and y axis in a rotated ellipse.
How can we find the maximum y?
2
votes
1answer
54 views
Finding eccentricity of an ellipse from latus rectum
The latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is the same as latus rectum of a parabola $y^2=4cx$ . Find eccentricity of the ellipse .
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1answer
24 views
Parabolic projectile equation demonstration question
I was looking at a book of physics and, it will sound dumb, but while I know that the maximum height equation of a projectile is max=(v·senα)/2g, I can't understand how do you get there from ...
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vote
3answers
57 views
Would a circle overlap a parabola's bottom by more than just its vertex?
I mean, out of the condition that a circle actually crosses the parabola.
My question is when a circle is "inside" a parabola, would it touch part of the parabola other than just the parabola's vertex ...
2
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0answers
39 views
How to find a point in an ellipse given the angle
I found a couple of formulas but I can't transform them in code.
From the answer in Calculating a Point that lies on an Ellipse given an Angle , for instance, I get to:
...
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vote
1answer
54 views
Find next point in ellipse given the chord length
I would like to draw a cloud programmatically. For this reason I need to know where to draw the next circle around the ellipse.
Given the chord (circle radius), how can I calculate the next point in ...
1
vote
1answer
62 views
Solve for an Ellipse Tangent to 2 Lines [duplicate]
I'm trying to automate creation of a curve in PowerPoint.
Here's an image of what I'm working towards:
I'm trying to show a diagram of a rocket trajectory from a launch site on Earth to a circular ...
1
vote
1answer
33 views
Find equations of the ellipses given conditions on the directrices, foci, and vertices
The ellipses have their centers at the origin and their major axes on the $x$-axis. Find the equation:
with distance between directrices $27$, and between foci $3$;
with a focus at $(-\sqrt{13},0)$ ...
1
vote
1answer
24 views
Find the equation of the hyperbola given foci and the minor axis
first time posting and using the site. I have a quick problem that I need some help with. I need to find the equation of a hyperbola given the foci and the length of the minor axis.
The foci ...
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vote
2answers
52 views
Angles and ellipse (proof)
First of all, sorry for my poor English! Can you please help me? I'm trying to prove that, given a point P at an ellipse.
Please help me prove that the angles are equal.
Thanks!
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votes
1answer
87 views
hyperbola: equation for tangent lines and normal lines
Find the equations for
(a) the tangent lines, and
(b) the normal lines,
to the hyperbola
$y^2/4 - x^2/2 = 1$ when $x = 4$.
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vote
2answers
45 views
conic sections, ellipse
A particle is travelling clockwise on the elliptical orbit given by
$$\displaystyle \frac{x^2}{100} + \frac{y^2}{25} = 1$$
The particle
leaves the orbit at the point $(-8, 3)$ and travels in a ...
1
vote
2answers
60 views
How do you find the distance between two points on a parabola
So I've been wanting to figure out a formula for an odd pattern I found... but to write a proof, I need to know one thing...
How do I find the distance between two points on a parabola?
Like, if I ...
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vote
2answers
37 views
Finding the tangents common to two rotated ellipses?
Is there a way to find the four tangents that two rotated ellipses share?
I believe that if two ellipses do not intersect and do not contain one another, they will have four tangents in common and I ...
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vote
2answers
75 views
How to calculate ellipse sector area *from a focus*
How do you calculate the area of a sector of an ellipse when the angle of the sector is drawn from one of the focii? In other words, how to find the area swept out by the true anomaly?
There are ...
5
votes
1answer
59 views
Determine if a conic is degenerate with the determinant.
There is a natural bijection between conics (written as homogeneous quadratic) and 3x3 matrices:
$$C=aX^2+2bXY+cY^2+2dXZ+2eYZ+fZ^2\Leftrightarrow \left(\begin{array}{ccc}
a&b&d\\
...
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vote
1answer
38 views
Car parking problem
I want to park my car doing similar to the one in the image. But I want to define a curve such that I park the car at once (without going forward, always backward). Suppose that the place that I want ...
3
votes
2answers
42 views
How do different definitions of ellipse translate to the same thing?
There are 2 definitions of an ellipse that I know.
One definition goes:
The locus of a point moving in a plane such that the ratio of its
distances from a fixed line (directrix) and a fixed ...
1
vote
1answer
29 views
Application of derivative - tangents to latus rectum
Drawn thru the focus of parabola is a chord perpendicular to the axis of the parabola. Two tangent lines are drawn through the points of intersection of the chord and the parabola. Prove that the ...
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3answers
70 views
How to write this conic equation in standard form?
$$x^2+y^2-16x-20y+100=0$$
Standard form? Circle or ellipse?
2
votes
1answer
21 views
How to prove and derive coefficients that an exponential whose base decreases exponentially is a parabola
Suppose I have a function defined by this recurrence-relation:
$$R(0) = r$$
$$R(n) = R(n-1) * (1+G)d^{n-1}$$
Here r is a base value, G is a base growth rate (G>0) and d is a decay in the growth rate ...
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votes
0answers
98 views
Conversion from Standard Ellipse Function to General Ellipse Function
I wonder if anyone can assist/show me how to complete this task...
I have the following equation which models a dual axis magnetic field:
$$\begin{equation} B_{H}^2 = B_x^2 + B_y^2 ...
2
votes
1answer
78 views
proof that intersection of two conic sections will intersect at at least two points.
In the following equation ρ(x,y) returns a constant value for a given coordinate.
n is the normal vector to the surface of the form [P,Q,-1] and s is a direction vector.
Using s = [Sx,Sy,Sz], the ...
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0answers
34 views
Representing conic sections as straight lines
Is there a projection that projects any conic section in a two dimensional orthogonal coordinate system with a focus at the origin into a potentially infinite set of parallel straight lines in a two ...
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1answer
43 views
Determine the Angle of an point in an Ellipse
I would like to know how to determine at which angle a point lies in an ellipse. Suppose I have an ellipse with semimajor and semiminor of 10 and 5 (see ...
3
votes
2answers
37 views
What is the rationale for the factor of $4$ in the Conics parabola equation?
The Conics form of a parabola equation is $4p(y-k)=(x-h)^2$ where $(h,k)$ is the vertex of the parabola and $p$ is the distance from the vertex to the focus. (Which is also the same distance from the ...
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vote
2answers
47 views
Computing the Semimajor and Semiminor axis of an Ellipse
I have the equation of the ellipse which is $\frac {x^2}{4r^2}+\frac{y^2}{r^2}=1$ Putting the (4,2) point on the ellipse we get $r^2=8$ so we get the equation $\frac {x^2}{32}+\frac {y^2}8=1$ and the ...
1
vote
1answer
52 views
Different curves
I stuck on a following question.
The curve is given by:
$(3-k)x^{2}+(7-k)y^{2}+9x+9y+7=0$ For which parameter $k$ k the curve will present
1)ellipse or circle
2)parabola
3)hyperbola
Thanks a lot!
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votes
2answers
101 views
Finding the Width and Height of Ellipse given an a point and angle
I have ellipse, lets say that the height is half of its width and the ellipse is parallel to x axis. then the lets say the center point is situated in the origin ...
2
votes
1answer
48 views
Minimum distance between $x = -y^2$ and $(0,-3)$
Find the minimum distance from the parabola $x + y^2 = 0$ (i.e. $x = -y^2$) to the point $(0,-3)$.
This is a homework question. When I try to use the derivative and substitute $-y^2$ for $x$, I ...
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vote
0answers
30 views
How can I find $\int{\sqrt{\left(b^2-1\right)x^2+1\over-x^2+1}}dx$?
I got this from the perimeter of an ellipse. I came up with the formula: arclength of f(x) for x from a to b=$\int_a^b\sqrt{f'(x)^2+1}dx$. Since an ellipse has the equation: $$\left({x-h\over ...
