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.

learn more… | top users | synonyms (3)

0
votes
3answers
244 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
votes
0answers
42 views

Point of intersection closest to the origin

How do I find the point of intersection of $𝑥 + 𝑦 - 𝑧 + 2 = 0$ and $𝑧^2 = 𝑥^2 + 𝑦^2$ that is closest to the origin? I know I have to use the LaGrange multiplier in order to minimize the ...
1
vote
1answer
65 views

Finding the conic section given equations of double cone and plane

Given the function of a double cone and a plane, how do we find the intersection between them? Suppose the equation of the cone is $f(x, y, z) = 0$ and the equation of the plane is $h(x, y, z) = 0$. ...
0
votes
3answers
109 views

Find the point on the parabola $2y=x^{2}$ that is closest to the point $(-4,1)$

The first part of the derivative which is to the power of $-1/2$ is too small to be considered relevant. I'm not sure how to proceed from here. The answer is $(-2,2)$ but I am not sure how to get ...
0
votes
4answers
82 views

Equation of circle touching a parabola

Suppose we have a parabola $y^2=4x$ . Now, how to write equation of circle touching parabola at $(4,4)$ and passing thru focus? I know that for this parabola focus will lie at $(1,0)$ so we may ...
3
votes
4answers
44 views

Given $\vec r(t)$, what are $\vec v(t), \ v(t), \ \vec a(t), \ a(t)$?

I have come to a problem that simply states that we have a parametric curve $$\vec r(t) = (2\sin t, 3\cos t), \ \ t\in \mathbb R$$ and asks that we find $\vec v(t), \ v(t), \ \vec a(t), \ a(t)$. ...
0
votes
1answer
33 views

Alternative proof of the reflection property

I'd like to prove the reflection property for the hyperbola. That is, that S'PS is bisected by the tangent at P. Suppose the tangent intersects the x axis at T. The usual method would be to use the ...
1
vote
1answer
131 views

Finding equation for diagonal ellipse given foci and eccentricity

Problem: Find the equation for the ellipse that has foci $$F_1 = (0, 0)$$ $$F_2 = (1,1)$$ and eccentricity $$\varepsilon = \frac12.$$ Hint: Use a rotation that moves the foci to the x-axis. My ...
1
vote
1answer
43 views

Determining conic section equation given foci and sum of distance to each point

Disclaimer: This title was hard to formulate. Edits welcome. Problem: Given foci $$F_1 = (1,0)$$ $$F_2 = (3,0)$$ of a conic section, find the equation for all points $P$ that satisfy $$|PF_1| + ...
0
votes
1answer
227 views

Finding the equation for a hyperbola given foci and eccentricity

Problem: Find the equation for the hyperbola which has foci $$F_1 = (-1, 3)$$ $$F_2 = (3,3) $$ and eccentricity $$\varepsilon = 2$$ Hint: Use a translation which moves the foci to the x-axis. My ...
3
votes
3answers
53 views

Finding the distance from ellipsoid to plane

I'm having problems with finding the distance from the ellipsoid $x^2+y^2+4z^2=4$ to the plane $x+y+z=6$. The question hinted that I'm supposed to find the distance from a point to the plane and ...
1
vote
1answer
64 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 ...
0
votes
1answer
120 views

How to find the sum of maximum and minimum curvature in an ellipse?

I am having difficulty finding the sum of maximum and minimum curvature of the ellipse $9(x-1)^2 + y^2 = 9$. I know that I am supposed to parametrize the ellipse as $f(x(t), y(t))$, with $x(t) = 1 + ...
0
votes
2answers
104 views

How does the eccentricity of a conic section define its shape?

Problem: Let $P$ be a point in the plane, $L$ a line containing $P$, and $\varepsilon$ a positive number. The triple $(\varepsilon, L, P)$ will then define a degenerate conic section. $\varepsilon$ ...
1
vote
1answer
305 views

How can I find the equation of a parabola only given it's x-intercepts?

I received a problem in my math class the other day that left me stumped. The problem went something like this. Mr. Lots-O-Cash would like to order a parabola that passes through the points $(-4, ...
1
vote
0answers
80 views

Finding eccentricity, directrix, foci of diagonal ellipse by rotating it

A problem I'm working on has the equation for a diagonal ellipse $$5x^2 + 5y^2 - 6xy - 8 = 0$$ which can be rotated 45 degrees to get the vertical ellipse $$\frac{x^2}{1}+\frac{y^2}{2^2} = 1$$ The ...
2
votes
2answers
31 views

Equation of tangent from a point outside it

Is there any general method of finding the equation of the tangent of a function $f(x)$ from a point $(a,b)$? $\hspace{1 mm}$ Then how do you find the angle between two tangents from $(0,0)$ to a ...
0
votes
3answers
80 views

Rotating a conic section to eliminate the $xy$ term

Problem: Given the equation $$5x^2 + 5y^2 - 6xy - 8 = 0$$ defining a non-degenerate conic section, find a rotation of the variables, such that the cross term $-6xy$ disappears in the new coordinates ...
1
vote
1answer
98 views

Locus of the centers of the circles tangent to a given line and circle

Say you are given a circle $C$ and a straight line $l$ exterior to the circle. How to describe the set of centers of circle that are tangent to both the $C$ and $l$? I have no idea how to proceed. My ...
0
votes
1answer
55 views

How to find the length of the focal chord making angle $\theta$ with the axis of parabola?

A focal chord of $Y^2 = 4aX$ makes angle $\theta$ with the axis of the parabola. How can I find the length of the chord? I have used the parametric equation but couldn't go further.
0
votes
2answers
134 views

How to derive the equation of a parabola from the directrix and focus

Could someone please offer me proof and explanation of the following? - I am just having trouble with finding the '$a$' part of the equation. "The leading coefficient '$a$' in the equation $$y−y_1 ...
0
votes
0answers
60 views

Fitting an ellipse such that the ratio of its radii is in a range

I need to fit an ellipse to a group of points. However, I have an issue and I appreciate if anyone can help me. The issue is that I need to have the fitted ellipse such that the ratio of its radii is ...
0
votes
1answer
47 views

Difficulties in understanding ellipse's minor axis's equation

I'm implementing an ellipse detector using some pdf I found on the internet, but I encounter some difficulties in understanding one of the equations. Here is the pdf: ...
2
votes
0answers
86 views

Finding equation for conic section given five points

Problem: Given the points $$(0,1),(0,-1),(2,0),(-2,0),(1,1)$$ find the equation for the conic section that passes through these points. My attempt: Using the general equation for a conic section, ...
1
vote
1answer
16 views

Reduction of general conic

The given equation is - $$3x^2 + 2xy + 3y^2 - 32y +92=0$$ To get rid of xy term i used the substitutions - $$x=p+q , y=q-p$$ Then the equation becomes - $$(p-4)^2 + 2(q-2)^2=1$$ which is an ellipse ...
0
votes
3answers
142 views

Construction of an ellipse

Is it possible to construct an ellipse with a line, compasses and a pencil? If yes, how and why is the construction correct?
1
vote
0answers
86 views

Finding the equation of a rational function or a conic section given three points

I have a rational equation derived from 2 points, $(2, 2)$ and $(10, 10)$. Solving for the rational equation gives the equation $$y = \frac{20}{12-x}.$$ What I want to happen right now is that given ...
1
vote
1answer
89 views

Super conic sections?

I know graphs of the form $A x^2 + B xy + C y^2 + D x + E y + F = 0$ are conic sections. But what would happen if I changed the highest power to 3? Would this be a new 3D shape, a 4D version of it, or ...
3
votes
1answer
111 views

Why are elliptic/parabolic/hyperbolic PDEs called “elliptic”/“parabolic”/“hyperbolic”?

I see that the form of a (e.g.) parabolic equation is $$Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0$$ with $B^2-4AC=0$ whereas the equation of a parabola is $$Ax^2 + 2Bxy + Cuy^2 + Dx + Ey + ...
2
votes
1answer
49 views

Sections of cones in higher dimensions

Everybody knows that when a plane intersects a cone at different angles and positions, we get conic sections. But, I wanted to know that if the same was possible in higher dimensions. If we take the 4 ...
0
votes
2answers
74 views

Conic sections directrix and focus

I do not understand the following: The equation of a particular parabola is: $$(y−​23)​​ = -\dfrac{1}{​16}\!\!\!\!\!​​​​(x+3)​^2​​$$ Given the equation of a parabola is - $(y−y_1)​= ...
2
votes
2answers
55 views

hyperbolic tangent vs tangent

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side (in a right triangle). Is there a similar definition for the hyperbolic tangent? The reason ...
0
votes
2answers
24 views

Show that the surface $x^2+y^2=x$ using $\theta \space and \space z$ can be parametrised by $(\cos^2(\theta), \cos(\theta) \sin(\theta), z)$

I really have no idea how to do this: $x^2-x+y^2=0$ looks like it can be a circle given by: $(x-\frac{1}{2})^2+y^2=\frac{3}{4}$ mostly $x=r\cos(\theta) \space and \space y=r\sin(\theta)$ work as ...
0
votes
1answer
75 views

Hyperbola / Rotated Hyperbola Intersection

I am trying to find the point where two hyperbolas intersect, that is, to find a vertex that is common to both hyperbolas. Also, note that I am only testing for a region of both hyperbolas -- only a ...
2
votes
4answers
81 views

Standard Form for a Parabola

What is the standard form for the following problem? I already know that it is a horizontal parabola. I just can't seem to be able to change it into the standard format. $8y² +96y-12x+240 = 0$ I ...
0
votes
1answer
34 views

Standard Form of Hyperbolas

If I have the equation $9x^2-4y^2-72x=0 $ and I know that is a hyperbola, how would I find the standard form for this equation? I'm not sure how to convert this equation to the standard form of a ...
1
vote
1answer
34 views

Conic Sections and Foci of Ellipses

We're just learning about ellipses and conics, and I'm a bit confused with ellipses, parabolas, circles, and hyperbolas, so a little help with this sample problem would be great. In which of the ...
2
votes
0answers
49 views

Partial Integral of an ellipse

this is my first question on stack exchange so please bear with me. I am trying to generate a synthetic image of an ellipse in Matlab where each pixel is shaded according to how much of that pixel ...
3
votes
0answers
35 views

Probability Density Function for Randomly Oriented Ellipse

I have an ellipse with a long aspect of a and a short aspect of b. The equation for this ellipse is found on this post: What is the general equation of the ellipse that is not in the origin and ...
0
votes
0answers
27 views

Non-standard 3D rotation of a set of points [duplicate]

I want to create a 3D surface as shown in the figure below. Toward this, I thought if I rotate a set of points in $xy$-plane on a elliptical arc I may be able to get such a surface. I was thinking of ...
1
vote
0answers
31 views

Why is this conic dual problem infeasible?

The problem is: $$\min \ x_2 : Ax -b = [x_1 \ 2x_2 \ x_1]^T \ge_{L^3} 0$$ where $L^m$ is the Lorentz cone. Which I found to have an optimal solution when $x_2 = 0$. I have shown that the conic ...
0
votes
1answer
29 views

What space curves can this theorem describe?

We were given the following theorem in our Vector Calculus class: THM: For space curve $R$ which does not pass through the origin, and which has a second derivative, the following are equivalent: ...
1
vote
1answer
37 views

Tangents of Rectangular hyperbola

P,Q,R are points on a rectangular hyperbola, and PQ perpendicular to PR. Prove that the tangent at P is perpendicular to QR.
4
votes
3answers
86 views

Generic rotation to remove Quadratic Cross-product

Show that if $b\neq 0$, then the cross-product term can be eliminated from the quadratic $ax^2 + 2bxy + cy^2$ by rotating the coordinate axes through an angle $\theta$ that satisfies the equation $$ ...
3
votes
1answer
202 views

Equivalence of geometric and algebraic definitions of conic sections

I have not been able to find a proof that the following definitions are equivalent anywhere, thought maybe someone could give me an idea: A parabola is defined geometrically as the intersection of a ...
0
votes
0answers
41 views

Find equidistant points along a mathematically known path/trajectory

Is there anyway to find the positions of points with equal distance from each other along a known shape? For example, I'd like to find equidistant points on an ellipse or a 2D eight curve the way ...
0
votes
1answer
36 views

Polar equations of circles and ellipses

I have been trying to convert some conic sections from rectangular to polar form. I am fine going the other direction (given polar, convert to rectangular), but am having trouble going the opposite ...
1
vote
1answer
38 views

How so I put these in Standard form? Circle, Ellipse or Hyperbola?

I need help putting these into standard form so I can graph them. Also need help figuring out which ones are which: $$25x^2-16y^2-150x+64y-239=0$$ $$9x^2+4y^2+54x-64y+301=0$$ $$x^2+y^2-6x+8y+3=0$$
4
votes
5answers
257 views

How to geometrically prove the focal property of ellipse?

How to prove geometrically that if we have a tangent of ellipse with focus F and F' in point P, that tangent is bisector of the angle between a line joining focus F to point P and the line F'P outside ...
0
votes
2answers
27 views

Explanation of graphical mathematical anomaly (for me, anyways)

I was working on some competition stuff when I came across the equation $y^2+2xy-x^2 = 0$, and the thing that surprised me was, when I graphed it, I got these two perpendicular lines at the origin, ...