1
vote
1answer
19 views

need explanation of what exactly is a directrix & focus?

((I'm not asking why do we need to know conic sections etc.) Like other similar questions.) I actually love math & currently learning conic sections in class, neither my textbook or teacher ...
2
votes
2answers
37 views

Orthogonal tangents to an ellipse

This is the problem I found back in the first year in the university. Suppose we have a non-degenerate (i.e. not a point and not an empty set) ellipse $E\subset \Bbb R^2$. Now define a set $D$ by a ...
3
votes
1answer
35 views

Locate a point a given distance from another point on an ellipse

Similar to Point on circumference a given distance from another point, but for an ellipse. Unfortunately, the difference is non-trivial. I have an ellipse and a point (C) that is somewhere on the ...
1
vote
1answer
43 views

Identify the locus.

Let $A,B,C$ lie on a straight line. $B$ is lying between $A$ and $C$. Consider all circles passing through $B$ and $C$. The point of contact of the tangents from $A$ to these circles lies on ..... We ...
1
vote
0answers
37 views

Equation of intersection of two cones

The equations of two cones are given; $(x-x_{0})^2+(y-y_{0})^2=\frac {(z-z_{0})^2}{m^2}$ and $(x-x_{1})^2+(y-y_{1})^2=\frac {(z-z_{1})^2}{m^2}$ How to find the equations of intersections 1) ...
2
votes
2answers
60 views

Why does the “T=0” method to calculate tangent work?

Given a random equation of a curve: $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$. Suppose we need to find the tangent to this curve at any point $A(x_1, y_1)$. A method given to me by my professor was the ...
0
votes
1answer
17 views

How do I find the width of a given section of an ellipse?

How would I be able to find the width of a horizontal ellipse (with a major axis of 120 and a minor axis of 5) at any given point along the major axis?
0
votes
0answers
27 views

Conic equation from cone/plane intersection

In an orthonormal cartesian frame $(O; \vec{x}, \vec{y}, \vec{z})$ consider: an infinite plane $P$ defined by: a point $p = (p_x, p_y, pz)$ an normal vector $\vec{n} = (n_x, n_y, n_z)$ a cone $C$ ...
4
votes
1answer
81 views

Minimum eccentricity of ellipses around another ellipse

Six circles can surround another circle of equal size, with each circle touching both the central circle and its two neighbouring outer circles. For sufficiently eccentric ellipses, it is possible to ...
1
vote
1answer
33 views

Problem in conics question

A vertical line passing through the point ($h$,0) intersects the ellipse $$\frac{x^2}{4}+\frac{y^2}{3}=1$$ at the points P & Q.Let the tangents to ellipse at P & Q meet at the point R.If ...
0
votes
1answer
54 views

Is this a correct way to derive the equation of an ellipse/hyperbola?

I was just testing to see if I could derive the equation of an ellipse (and consequently a hyperbola) with the least amount of information to remember. The small amount of information I chose to use ...
0
votes
1answer
62 views

Ellipse Tangents in 3D

I know that we can find the tangent of the ellipse in 2D by taking the derivative of the equation defining the ellipse. But I'm little bit confused about finding the ellipse tangent in 3D. Where the ...
0
votes
1answer
31 views

Homography between ellipses

This is a spin-off from a comment on Stack Overflow. How can I find a homography between two ellipses in the plane?
0
votes
0answers
21 views

How are the sine functions along with the hyperbolic functions visualized with imaginary rotations?

Since we know that: cos(t)=cosh(it) and isin(t)=sinh(it) I've been thinking about this, and obviously this is referring to how if you move at a right angle from a circle on a conic section, you end ...
5
votes
2answers
121 views

Area under parabola using geometry

We have to find the area of the pink region. As we all know this can be evaluated using limiting its Riemann sum, of which its a standard example. However I want to know if this can be done without ...
1
vote
3answers
82 views

General form of a circle

My math teacher taught me that the general form (equation) of a circle is: $$ ax^2+by^2+cx+dy+e=0 $$ He also asked us this: If the product of $c$ and $d$ is negative, then what 2 quadrants can the ...
4
votes
5answers
79 views

The equation of parabolas.

I have trouble grasping some basic things about parabolas. (This should be easily found on Google, but for some reason I couldn't find an answer that helped me). I know one simple standard equation ...
1
vote
1answer
69 views

Calculate Ellipse From Points?

How can I calculate an ellipse from a group of points ? Result: center point, x-radius, y-radius ? I'm not mathematician so I don't really know the best parameter style for ellipses. This ellipse ...
0
votes
1answer
61 views

Intersection of conics

By conic we understand a conic on the projective plane $\mathbb{P}_2=\mathbb{P}(V)$, where $V$ is $3$-dimensional. I'd like to ask how to find the number of points in the intersection of two given ...
0
votes
1answer
79 views

Proving properties of an ellipse

I'm studying about ellipse and its properties. My reference is the following pdf: http://nebula.deanza.edu/~bloom/math43/ellipse-derivation.pdf My questions are from the very first page of the ...
0
votes
2answers
159 views

Length of chord on ellipse

Suppose I have an ellipse centered at the origin, preferably expressed in its matrix form, and I want to know the chord length of a segment that passes through the origin with the endpoints at the ...
0
votes
2answers
222 views

Find angle at given points in Ellipse

I have Ellipse's center-points, minor-radius and major-radius. I can find, how to check if given point(x, y) exists in Ellipse or not. Now, I want to find given point(x,y) exists at which angle in ...
1
vote
1answer
29 views

Foci Concentric Circles

My approach: Using the foci formula $$c=\sqrt{a^2-b^2}$$. By plugging in $a=3$ and $b=2$ I obtain plus and minus $\sqrt{5}$. But there's 2 choices with a root 5 result. How do i know which one is ...
1
vote
1answer
88 views

Reflection inside an ellipse

From a typical point $P$ inside an ellipse, how many points $Q_i$ on the ellipse have $PQ_i$ normal to the ellipse? Someone asked me at school many years ago but I don't think I worked it out.
3
votes
0answers
86 views

Focus-Focus Definition for a Parabola

I am looking at the focus-focus definitions of the conics, i.e. defining them as the locus of points with the property that some function of the distance from the point to two foci is a constant. ...
0
votes
1answer
52 views

Question about finding a third point on an ellipse given angle

If I have a known point $Y$ on an ellipse in the first quadrant, and known point $X$ on the $x$-axis, and some angle $\theta$ between $XY$ and $YZ$ with $Z$ being some mystery third point on the ...
1
vote
0answers
68 views

How to see and proof that the hyperbola as a constant difference of distances holds for $\frac{1}{x}$?

I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is $2a$, which is the distance ...
0
votes
0answers
30 views

can an ellipse be described by two circles?

I was intrigued by the fourier visualization and the simpsons face fourier and wanted to know the answer to the question can an ellipse be described by two circles ? and will the fourier analysis ...
1
vote
2answers
121 views

Creating an ellipse passing through a rectangle's vertices coordinates

Given a rectangle with vertices $A(x_1, y_1), B(x_2, y_2), C(x_3, y_3)$ and $D(x_4, y_4)$, how to create an ellipse with this vertices coordinates?
0
votes
2answers
258 views

Find equation of the circular cross section of a unit sphere

I have a unit sphere in Cartesian coordinates: $x^2 + y^2 + z^2 = 1$ or in spherical coordinates: $x = \rho \sin(\phi) \cos(\theta)\\ y = \rho \sin(\phi) \sin(\theta)\\ z = \rho \cos(\phi)$ I ...
0
votes
0answers
43 views

Quadrature of the parabole

This exercise is from a course in mathematics history. Find U: S: V, where S is the area of ​​a parabolic segment, U is the area of the largest triangle that can fit inside the parabola segment ...
2
votes
1answer
112 views

Why are two definitions of ellipses equivalent?

In classical geometry an ellipse is usually defined as the locus of points in the plane such that the distances from each point to the two foci have a given sum. When we speak of an ellipse ...
0
votes
3answers
72 views

parametrise equation of a hyperbola

Any point on an ellipse can be wrttien as $(a\cos\theta,b\sin\theta)$, How could we genarilse this to a hyperbola?
2
votes
0answers
28 views

What are defintions of ellipticity?

Ellipticity seems to have many definitions. So far, I'm aware of three. Are there other definitions of ellipticity that you know of, and if so, where did you encounter them? The three definitions ...
2
votes
6answers
215 views

Parabola in parametric form

Show that the following system of parametric equations describes a line or a parabola: $$\begin{cases} x=a_1t^2+b_1t+c_1 \\ y=a_2t^2+b_2t+c_2 \end{cases}, t\in\mathbb{R}.$$
0
votes
1answer
186 views

Expression for hyperbola on complex plane

The hyperbola $$x^2 - y^2 = 1$$ has a simple expression in the complex plane as $\{z^2 + \bar{z}^2 = 2\}$. Is there a similarly simple expression for a hyperbola ...
0
votes
3answers
69 views

Finding the second radius of an ellipse given the first radius and the center

I know the coordinates of A, B and C. A and B are on the axis L1. From that information, I can find the coordinates of the center, the length of radius r1 and the equation of L1 (see picture). Then I ...
1
vote
2answers
119 views

pick point on parabola so 2 conditions are true

You have a parabola $$y=ax^2+bx+c $$ We know $a,b$ & $c$. On this parabola you have to pick a point A where the following conditions are true: 1) If you draw a tangent line in this ...
0
votes
1answer
200 views

Hyperbola in polar coordinates, what's wrong?

I read that the equation of a conic in polar coordinates is $$r=\frac{l}{1+e\cos \theta}.$$ But when I try to reduce the hyperbola $$x^2 - y^2 =1$$ to that form by setting $x=r\cos \theta $, $y=r ...
3
votes
0answers
51 views

Biggest ellipse included in a convex polygon

Considering a N edges convex 2D polygon called P. Let's name its vertices $\{p_1, p_2, ..., p_N\}$ described in a counter-clockwise order, with $p_i = (x_i, y_i)$ What would be, and how would one ...
1
vote
0answers
40 views

Why loci of conic sections are defined in the way they are?

I understand how conic sections are produced i.e. when a plane cuts a double nappe right circular cone at different angles, we get different types of conic sections like parabola, ellipse etc. But I ...
1
vote
0answers
88 views

Plotting an elliptical arc given 3 points, radius ratio and angle

I'm trying to plot an elliptical arc. I know the starting point $P_1$, ending point $P_2$ and a control point $P_3$. I'm also given the ratio of radii $a/b$ and the angle $\theta$ of the ellipse. As ...
1
vote
1answer
695 views

Throwing a Projectile in 2D space

Normally I am strong at maths, but here I have a Math question that after spending 5 pages, I just can't figure it out. Here goes: A person is playing a game that requires throwing an object onto a ...
0
votes
3answers
132 views

Finding the equation of a circle ?

My Approach: I know that the general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c=0$. So, the aim is to fond the constants g,f,c.So, I should make equations relating these constants from the ...
6
votes
5answers
231 views

Parabolas through three points

We can draw an infinite number of parabolas that pass through three given points $A$, $B$, $C$ (in that order). For each such parabola, we take the tangent lines at $A$ and $C$, and intersect them to ...
1
vote
4answers
256 views

Parabola and Circle problem : The parabola $y =x^2-8x+15$ cuts the x axis at P and Q. A circle is drawn …

Problem : The parabola $y=x^2-8x+15$ cuts the x axis at P and Q. A circle is drawn through P and Q so that the origin is outside it. Find the length at a tangent to the circle from O. My approach ...
2
votes
2answers
75 views

Can a cyclic quadrilateral be inscribed in a parabola?

It is quite obvious that a quadrilateral can be inscribed in a parabola. However, can somebody provide a nice (meaning: intuitive) proof that a cyclic quadrilateral can be inscribed in it? Further, ...
2
votes
0answers
116 views

How to calculate a PHI-ellipse defined by 3 points and its width/length ratio

in the field of technical analysis for stock markets, the usage of so-called Phi-Ellipses is getting popular. One important property of this ellipses is its constant length/width ratio (e.g. 1.618). ...
0
votes
0answers
24 views

How to find start and sweep angle of the arc in a ellipse [duplicate]

i have Arc of the ellipse with start and end point of the arc and minor axis and major axis and rotational angle of ellipse , sweep direction of arc, and also if the arc is large or small . by ...
2
votes
0answers
124 views

Equation of an intersection of two cones when the intersection is an ellipse

The two cones with vertex $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ and generating angle of two cones is $\alpha$ given. I need to write the equation of the intersection of two cones ...