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.
0
votes
1answer
21 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 ...
1
vote
3answers
48 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
votes
0answers
30 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:
...
1
vote
1answer
40 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
53 views
Solve for an Ellipse Tangent to 2 Lines
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
29 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
16 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 ...
1
vote
2answers
45 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!
0
votes
1answer
65 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 $x = 4$.
1
vote
2answers
40 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
44 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 ...
0
votes
2answers
34 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 ...
1
vote
2answers
54 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
44 views
Determine if a conic is degenerate with the determinant.
There is a natural bijection between conics (written as homogeneous quadratic) and 3x3 matrices:
$$C=aX^2+2bXY+cY^2+2dXZ+2eYZ+fZ^2\Leftrightarrow \left(\begin{array}{ccc}
a&b&d\\
...
1
vote
1answer
34 views
Car parking problem
I want to park my car doing similar to the one in the image. But I want to define a curve such that I park the car at once (without going forward, always backward). Suppose that the place that I want ...
3
votes
2answers
37 views
How do different definitions of ellipse translate to the same thing?
There are 2 definitions of an ellipse that I know.
One definition goes:
The locus of a point moving in a plane such that the ratio of its
distances from a fixed line (directrix) and a fixed ...
1
vote
1answer
23 views
Application of derivative - tangents to latus rectum
Drawn thru the focus of parabola is a chord perpendicular to the axis of the parabola. Two tangent lines are drawn through the points of intersection of the chord and the parabola. Prove that the ...
0
votes
3answers
57 views
How to write this conic equation in standard form?
$$x^2+y^2-16x-20y+100=0$$
Standard form? Circle or ellipse?
2
votes
1answer
20 views
How to prove and derive coefficients that an exponential whose base decreases exponentially is a parabola
Suppose I have a function defined by this recurrence-relation:
$$R(0) = r$$
$$R(n) = R(n-1) * (1+G)d^{n-1}$$
Here r is a base value, G is a base growth rate (G>0) and d is a decay in the growth rate ...
0
votes
0answers
90 views
Conversion from Standard Ellipse Function to General Ellipse Function
I wonder if anyone can assist/show me how to complete this task...
I have the following equation which models a dual axis magnetic field:
$$\begin{equation} B_{H}^2 = B_x^2 + B_y^2 ...
2
votes
1answer
70 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 ...
0
votes
0answers
30 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 ...
0
votes
1answer
30 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
36 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 ...
1
vote
2answers
40 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
51 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!
0
votes
2answers
78 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
43 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 ...
1
vote
0answers
29 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 ...
1
vote
1answer
86 views
Incomplete elliptic integral of the second kind and the arc length of an ellipse - does a `simple` relation exist?
Short introduction
For a calculation I am working on I need to determine the arc length $l$ of a part of an ellipse in terms of the major axis $2a$, the minor axis $2b$ and the angle $\phi$.
I ...
2
votes
1answer
107 views
How do we know $\pi$ is a constant? [duplicate]
How did the ancient Greeks discover that the ratio of a circle's circumference to its diameter is constant? It does not seem so intuitive. Thanks!
1
vote
1answer
44 views
Turning an ellipse into a parabola
Today I was discussing circles, ellipses, hyperbolas, and parabolas in my precalculus class. We did the usual: completing the square, finding the center and radius (radii), etc. etc. But I like to ...
13
votes
8answers
2k views
What is the real life use of hyperbola? [closed]
I was doing hyperbola ,I was thinking does it have any real life uses or it just a mathematics theory?
1
vote
2answers
61 views
Good books on conic section.
Can anybody suggest me good books for conics section.I want it for IIT-JEE mains and advanced and also for ISC. It should be available in India .
0
votes
1answer
49 views
Finding a,b of elipse
Given $x^{2}+y^{2}=R^{2}$, so that we multiply every $x$ by $a$ and every $y$ by $b$, $(a>b)$
And the distance between the focuses of this locus is $48R$, and the area of the rhombus which ...
1
vote
1answer
113 views
How to find equation of parabola when we only know the equation of latus rectum and coordinates of vertex?
Suppose the equation of latus rectum is x=4 and the vertex is (2,3). I am confused wouldn't there be many parabola with this same vertex and latus rectum.If not how to find the equation?
The answer ...
0
votes
0answers
17 views
Number of solutions to a conic in $Z_p$
$p\not=2$ is a prime. $a, b, c, d\in F_p$ and $acd\not =0$. C is the conic given by the homogeneous equation
$ax^2+bxy+cy^2=dz^2$.
If $b^2\not =4ac$, prove that #$C(F_p)=p+1$
Note: Solutions are to ...
1
vote
0answers
29 views
Equation of a general conic from 3 points and the major axis
I have read that given 3 points on a conic and the equation ($ax+by+c=0$) of its major axis, we can write the equation of the conic ($Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$). I've seen it done by ...
0
votes
0answers
38 views
Parabolic segment problem
I have a problem. I have tried to solve it but I get $125/6$ instead of $9/2$ (textbook result).
Find the area of the parts of plane given by the solutions of the following system:
$$ ...
2
votes
1answer
56 views
Number of points determining a Quadric
I know that in $\mathbb{R}^2$ that 5 points in general linear position determine a unique conic (also non-degenerate). I was wondering about the higher dimensional analogue of this. Is it true, for ...
2
votes
1answer
76 views
Deriving hyperbola equation: why can we assume vertices lie in between foci?
I'm reading through a derivation of the standard equation of a horizontal hyperbola, and while I can follow the the algebra, I'm hung up on an assumption it makes early on: that the vertices lie in ...
2
votes
1answer
60 views
Integer solutions to a hyperbola
Is there a way to find all integer solutions to a hyperbola equation? If it helps, I am specifically looking at "square" hyperbolas (i.e. of the form $\frac{x^2}{z} - \frac{y^2}{z}=1$), where z is an ...
1
vote
2answers
43 views
Tangent line to a general conic at a point
If I have a conic
$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$
and want to know the tangent line at $(x_0, y_0)$, I thought I would just find the derivative (implicit) and use the equation of a line ...
2
votes
0answers
18 views
Do both these ellipses satisfy the same conditions?
I was solving a problem that asked for the equation of the ellipse with the following properties: vertex at $(-10,5)$, focus at $(-2,5)$, eccentricity $\frac{1}{2}$. I think I found two such ellipses, ...
0
votes
0answers
32 views
Representing an imperfect ellipse in 2 linear variables
I have several shapes which are roughly elliptical. I know the major and minor axes and the true circumference, so I store them like this:
$$a={\text{axis}}_{\text{major}}\\
...
2
votes
1answer
74 views
Find enpoints of major axis of an arbitrary ellipse using its general equation
I have a general equation of an ellipse in the form of
$Ax^2+Bxy+Cy^2+Dx+Ey+f=0$. How do I find the equation of (or even endpoints would work) major axis of an ellipse. I am aware of following ...
2
votes
0answers
25 views
How does this method to find the centre work?
Say we have a conic with equation $f(x,y)=c$.
My teacher says that it's centre satisfies the equations :
$f_x(x,y)=f_y(x,y)=0$ (If it has a centre).
She didn't give any explanation. I thought this ...
3
votes
1answer
59 views
Centers of the osculating circles along an ellipse
Consider an ellipse on the plane $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. We will use the usual parametrization: $P(t)=(x(t),y(t))=(a\cos t,b\sin t)$.
Then the tangent vector is $T(t)=(-a\sin t, b\cos ...
1
vote
0answers
154 views
Finding the eccentricity/focus/directrix of ellipses and hyperbolas under some rotation
Suppose I have an ellipse/hyperbola rotated about the origin by some angle $\theta$. Am I right in saying that the following general process will find the eccentricity $e$ of these conics?
Find ...
0
votes
2answers
38 views
Write the equation of an ellipse
The information given is the focus at (-2,3), directrix y=0 and eccentricity =1/2
