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)

3
votes
3answers
79 views

How to show that any rectangle in ellipse must be oriented parallel to axes?

A problem which is often given as an exercise for students learning about calculus and finding extrema, is to find maximal possible area of a rectangle inside an ellipse. Such question was asked, for ...
1
vote
2answers
46 views

Conic Sections: Hyperbola (Finding the Locus)

This is a multipart question so bear with me until I get to the part where I am stuck on. $H$: $xy=c^2$ is a hyperbola (i) Show that $H$ can be represented by the parametric ...
0
votes
0answers
28 views

Rotate an Ellipse

$x = h + a \cos(φ) \cos(θ) + b \sin(φ) \sin(θ)$ $y = k + b \cos(φ) \sin(θ) - a \sin(φ) \cos(θ)$ Hi, I have basic question of parametric equation for ellipse. I'm trying to rotate horizontal ellipse ...
0
votes
0answers
37 views

Finding $x^2$ and $y^2$ of hyperbola

Currently, I am trying to the $x^2$ and $y^2$ of a hyperbola. I have the vertices at $(-1, -1)$ $(5, -1)$ I have the focus at $(-4, -1)$ $(8, -1)$ I know that the distance between two vertices ...
6
votes
0answers
93 views

Why does the Ellipsograph/Trammel of Archimedes draw an ellipse, really?

Here's a diagram of the device I mean, hard at work drawing an ellipse. I find this quite surprising, and would like to get to the bottom of things. Essentially, a rod (black line in animation) is ...
10
votes
6answers
713 views

538.com's Puzzle of the Overflowing Martini Glass - How to compute the minor and major axis of an elliptical cross-section of a cone

FiveThirtyEight.com Riddler Puzzle / May 13 The puzzle goes like this; "It’s Friday. You’ve kicked your feet up and have drunk enough of your martini that, when the conical glass (🍸) is upright, the ...
1
vote
3answers
47 views

Same perimeter and area for a circle and an ellipse

For a given circle, is there exist an ellipse with same perimeter and area as to that circle? If not, that is my suspicion, is in three-dimension parallel question: For a given sphere, is there ...
1
vote
1answer
42 views

Identification of a conic section

Consider the equation $(E)\hskip 5mm x^2+xy+ky^2+6x+10=0$. I am looking for conditions on $k$ for the graph of $(E)$ to be a circle or an ellipse. Clearly, if it is a circle or an ellipse, its ...
0
votes
3answers
36 views

Equation of parabola given 2 points $(x_1,y_1)$ and $(x_2,y_2)$ in expanded form

I need to find an equation for the parabola that passes through the points $(0,0)$ and $(5,0)$, such that $f(x)<0$ whenever $0< x <5$. The answer should be in expanded form. I.e., $f(x)=ax^2+...
68
votes
2answers
803 views

Modelling the “Moving Sofa”

I believe that many of you know about the moving sofa problem; if not you can find the description of the problem here. In this question I am going to rotate the L shaped hall instead of moving a ...
1
vote
2answers
45 views

Points $P_i$ on an ellipse such that angle $P_iOP_{i+1}=\frac{\pi}{n}$

Consider an ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ with $O$ as the origin. $n$ points denoted as $P_1,P_2,\cdots$ are taken on the ellipse such that angle $P_iOP_{i+1}=\frac{\pi}{n}$ where $i\in(1,n-...
2
votes
3answers
92 views

4-ellipse with distance R from four foci

I'm trying to find the equation for the generalization of an ellipse called a $n$-ellipse which has a constant distance R from four foci located at $(0,0),(0,1),(1,0),(1,1)$ Edit: As an algebraic ...
0
votes
1answer
39 views

How do parametric equations work?

I was given a graph like this in my exam. Its defined para-metrically by x=c^2 and y =c^3. It won't help me now but could someone explain this to me why I have two seemingly different lines I know it ...
2
votes
2answers
29 views

Slope of axes of a General Conic Section

A General Conic Section is given by the equation $ax^2 + by^2 + 2hxy +2gx +2fy + c =0 $. Let the $\theta$ be the slope of one of its axes. Prove that : $$\tan 2\theta = \dfrac{2h}{a-b}$$ ...
0
votes
1answer
21 views

Hyperbolas and Quadrants on Rotation

Let's assume we have a standard hyperbola. On rotating the hyperbola $45^{\circ}$ clockwise, the new hyperbola should lie in the $2$nd and $4$th quadrant. However, the equation of a parabola rotated $...
5
votes
2answers
47 views

Why are there only two tangents to a hyperbola from a point, instead of four?

Why are only two tangents possible to a hyperbola from a point? If we treat the hyperbola as two individual parabolas, then a point should be able to create two tangents through it for both of them, ...
0
votes
0answers
13 views

ratio of areas of a cone and plane intersection after the plane is rotated

I have a right cone with the tip $V$ and tip angle $\alpha$, with the axis $d_1$ that it is orthogonal on the plane $P$ at distance $L$ from $V$. In the plane $P$ I have a line $d_2$ that intersects $...
-1
votes
1answer
73 views

Find the equation of the ellipse

An ellipse with centre at $(4,3)$ touches $x$-axis at $(0,0)$. If the slope of the major axis of ellipse is 1, then find the equation of the ellipse?
2
votes
1answer
36 views

Prove that the group of the rational points on the conic $u^2-Av^2=1$ is not finitely generated.

This is an exercise from Rational Points on Elliptic Curves by Silverman. Let $H$ be the conic $u^2-Av^2=1$ where $\sqrt{A}\notin \mathbb{Q}$. If $(u_1,v_1), (u_2,v_2)$ are two points in $H(\...
1
vote
1answer
24 views

Finding Coordinate along Ellipse Perimeter with Arbitrary Origin Coordinates

This is heavily related to: This Question I know that question should have handed me the answer, but I can't quite wrap my head around what I need to do to get coordinates with an arbitrary origin. ...
0
votes
1answer
12 views

Points on parabola with abscissa in A.P. and ordinate in G.P.

The points with coordinates $(a,b),(a_1,b_1),(a_2,b_2)$ are points on parabola $y=3x^2$. The numbers $a,a_1,a_2$ are in Arithmetic progression while $b,b_1,b_2$ are in Geometric Progression. Calculate ...
0
votes
1answer
12 views

Show that conic C has rank 2

Given the conic $C=lm^T+ml^T$ defined by two distinct lines $l$ and $m$. $C$ is a symmetric 3x3 matrix. How can i show that $C$ has rank 2? The rank of a matrix $M$ can be calculated using gaussian ...
2
votes
0answers
33 views

Fourier transform of function defining half an ellipse

I'm trying to determine the expression for the Fourier transform of a function defining half an ellipse. It's been awhile since I've done Fourier transforms by hand. Obviously I can plug the ...
0
votes
0answers
13 views

Defining Conic Section From 5 Points (No Matrix work)

I am doing a homework assignment, and I need to find: a) what type of conic it is b) the equation for the conic from $5$ points. In class we haven't done any work with matrices, so I need a method ...
9
votes
1answer
273 views

Right triangle on an ellipse, find the area

Beginning note: Please wait until the animations load. The loading might take some time depending on your internet connection. Secondly, the title and the content of the question might not be well ...
0
votes
2answers
14 views

How to find the solutions for the quadratic equation for conic sections $\epsilon \in (0,1)$

Going from this definition of the conic section: $\epsilon |Pl| =|PB|$, you get the following equation for the intersection with the $x$-axis: $y^2 = (\epsilon ^2-1)x^2+(B-\epsilon ^2L)2x+\epsilon ^2L^...
0
votes
0answers
7 views

Find a point on an ellipse given a different point on the ellipse and either an arc length or a chord length.

Given a point $A$ on an ellipse and an arc length, how can I find the resultant point $B$ on the ellipse such that point $B$ is the arc length away from Point $A$? Alternatively, given a point $A$ on ...
0
votes
0answers
44 views

Double Integral over the region of an ellipse cut off by a circle

I've been stuck on this question for awhile. I need to calculate the double integral $\iint_R \frac{1}{r^3} dA$ using polar coordinates. R is the region displayed below: The ellipse has centre (4,...
1
vote
0answers
15 views

Ellipsoid Axis at density contour, why choose biggest eigen value for axis?

I've been trying to figure out how to find the density contour for a multivariate normal density function with an arbitrary number of dimensions. I've found a lot of examples for 3Dimensions and for ...
0
votes
1answer
25 views

Find cone-plane intersection points in a construction

I have two points on the X axis, A and B, which are connected to the two points, C and D on the sketch plane parallel to XY plane. I have a point E which lies at distance h from D point in Y direction ...
1
vote
2answers
33 views

Prove that the directrix is tangent to the circles that are drawn on a focal chord of a parabola as diameter.

Question: Prove that the directrix is tangent to the circles that are drawn on a focal chord of a parabola as diameter. Here is a picture; What I have attempted; Let the parabola ...
1
vote
0answers
32 views

Find the equation of a hyperbola, given a point on it and the length of the transverse axis

My textbook has the following question: The transverse axis of a hyperbola is of length $24$ and the curve passes through the point $(13, 10)$. Find the equation of the hyperbola. Also give the ...
0
votes
2answers
71 views

How to get the properties of an ellipse with six points given.

I am looking for a way to calculate the lengths of both semi-axes and the rotation angle of the ellipse in the image as shown in this picture. Six points are given, with two pairs of points being ...
0
votes
0answers
18 views

Point halfway around ellipse quadrant

I want to find the length between the centre of an ellipse and a point, P, on the ellipse, where the arc length between P and the intersection of the semi-minor axis with the ellipse is equal to the ...
0
votes
0answers
13 views

Derive the parametric form of the locus of point where difference between distance to two points is constant

Given two points $P_1=(x_1,y_1)$ and $P_2 = (x_2,y_2)$, the locus of the point whose (signed) difference between the distance to the two points is a constant $\Delta$ is one branch of a hyperbola ...
1
vote
0answers
27 views

Hyperbola equation proof

I've been trying to prove the canonical form of the hyperbola by myself. $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ I started from the statement that ...
0
votes
1answer
37 views

How to derive formula for focus of a parabola?

I understand how to obtain the formula for the vertex of a formula, $ y= a(x-h) + k $ where $ h=-b/2a$ and the vertex is $(h,k)$. However I don't know how to get to $(h,k+1/4a)$. Could someone please ...
0
votes
2answers
57 views

Ellipse - relation between a and b such that $F_1P \perp F_2P$

Consider the ellipse $\displaystyle \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}=1$ with foci $F_1 (-e, 0)$ and $F_2 (e, 0)$ (where $e$ is the linear eccentricity). What is the relation between $a$ and $b$ so ...
0
votes
2answers
35 views

How to determine standard equation of a conic from the general second degree equation?

From a given general equation of second degree i can determine the conic by following rules: Given equation: $ax^2+by^2+2hxy+2gx+2fy+c=0$ then if, $abc+2fgh-af^2-bg^2-ch^2$ is not equal to zero ...
0
votes
0answers
56 views

Computing the properties of the 3D-projection of an ellipse.

I have an ellipse that is rotated around the white axis (see image below) in 3-dimensional space by an angle α. The axis passes through the perimeter and one of the ...
0
votes
1answer
19 views

Ellipse from two arbitrary points, tangent at P1 and a focal point

Is it possible to find this? Really only need the semi major axis or even it's orientation. Please see the linked image. Known elements are in red and the desired element is in blue.
2
votes
2answers
34 views

Finding the tangent of an ellipse that is perpendicular to a line

The books say's "Find the equations of the tangents to $x^2+3y^2=4$ which are perpendicular to the line $x-2y=7$" I've graphed them and found that the given line does not pass through the ...
0
votes
1answer
40 views

circle cuts three circles at the extremities of the diameter

If the circle $$x^2 + y^2 + 2gx + 2fy + c = 0$$ cuts the three circles $$x^2 + y^2 – 5 = 0\space;\space x^2 + y^2 – 8x – 6y + 10 = 0 \space;\space x^2 + y^2 – 4x + 2y – 2 = 0;$$ at the ...
1
vote
3answers
30 views

Algorithm: Intersection of two conics

I am looking for a detailed description of an algorithm for the classical problem of computing the intersection of two conic curves. The curves are given by two equations of the form: $$ a x^2 + b y^...
0
votes
0answers
19 views

$(x-1)(y-2)=5$ and $(x-1)^2+(y+2)^2=r^2$ intersect at four points $A,B,C,D$. Centroid of $\Delta ABC$ lies on $y=3x-4$, then the locus of $D$

$(x-1)(y-2)=5$ and $(x-1)^2+(y+2)^2=r^2$ intersect at four points $A,B,C,D$. If centroid of $\Delta ABC$ lies on $y=3x-4$, then what is the locus of $D$? I did try a couple of things, but I honestly ...
0
votes
0answers
37 views

Volume of paraboloid that is cut with plane

How to calculate the volume of the paraboloid : x^2 + y^2 = z that is cut with x + y + z = 5 plane. Please give several methods if you can. Thank you very much for answers.
0
votes
1answer
22 views

Condition for common tangents to Circle and parabola

The parabola $y^2=4ax$ and circle $x^2+y^2+2bx=0$ have more then one common tangents;, Then which one is/are right, $(a)\; ab>0\;\;\;\; (b)\; ab<0\;\;\;\; (c)\; ab<-2\;\;\;\; (d)\; ...
0
votes
1answer
15 views

Equation of tangent at vertex

The equation of tangent at vertex of parabola $4y^2+6x=8y+7$ is .. I simplified the equation and got $4(y-1)^2 =-(6x-11)$. What do I do further?
1
vote
1answer
19 views

Focus of a parabola

If (2,0) is the vertex and y-axis the directrix of a parabola find the focus of the parabola. What does y-axis is directrix mean here?
1
vote
2answers
27 views

Partial differentiation and tangency

$\text{Q}$ Write the equation of tangent at the vertex of the parabola $2y^2+3y+4x-3=0$ . $\text{My attempt}$ if I partially differentiate the curve with y then I get $y+0.75=0$ which is correct ...