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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18 views

orthogonal diagonalization to sketch equation: $5x^2-24xy-5x=13$

to sketch this i wrote the equation down in the form: $X^TAX=13$ where $X^T=[x\:\:y]$ and $A=\begin{bmatrix} 5&-12\\ -12&-5 \end{bmatrix} $. Then, $A$ is orthogonally diagonalizable so there ...
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37 views

Prove an ellipse is unique if the foci and a tangent are given.

Given 2 points $F_1$ and $F_2$ and a straight line $l$ which does not cross $[F_1F_2]$. Prove that there exists an unique ellips with $F_1, F_2$ as foci, and tangent $l$. What if $l$ crosses ...
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1answer
28 views

Finding the condition for point of intersection of three normals to a given parabola

Question: Suppose that the normals at three different points on the parabola $y^2=4x$ pass through the point (h,0). Show that h>2. My attempt: Equation of normal to parabola $y^2=4x$: ...
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1answer
22 views

Points of intersection of vector with cone.

I have a Vector $\vec A$ defined as : $(A_o+t*A_d)$ I also have a Cone with vertex (cone tip) V and axis direction $\vec D$, base radius R and height H. The cone angle can be computed via ...
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14 views

How to find smallest tangent ellipse from multiple lines?

The ellipse I'm trying to calculate must be tangent to at least four lines. This ellipse must also intersect the other lines. I've tried using quadrilaterals and by transforming the equation of these ...
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38 views

Hyperbola and 3 normals from point P

From any point P on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ three normals other than that at P are drawn. Find the locus of the centroid of the triangle formed by feet of the normals. Do we ...
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31 views

Is it actually important to translate the conics?

I've been studying conics and I'm curious about one aspect. There is a common way of doing it. Taking the quadratic form, we make a rotation and a translation and then the cross-term and the linear ...
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1answer
9 views

Eigenvalues order in conics reduction?

I'm doing exercises on the reductions of conics to canonic forms using eigenvalues. I'm trying to understand what does actually change when I put the two eigenvalues that I find in a different order ...
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44 views

Inversion across an ellipse

Let's take an ellipse with the standard equation $$\frac{x^2}{9}+\frac{y^2}{4}=1$$ And I am trying to invert the following ellipse across that ellipse ...
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1answer
67 views

Locus of centres of circles tangent to two fixed circles?

Find the locus of the centres of circles tangent to two fixed circles. From my initial observations, I strongly think that the locus may be part of a hyperbola or some other conic? (because the ...
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0answers
36 views

Inversion across a general ellipse

This paper is very useful in how it explains the mapping of any coordinates $(x,y)$ across an ellipse with the function $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ to ...
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3answers
32 views

Check if disk lies within an ellipse

I have an ellipse in normal form centered at the origin and want to check whether a disk with given center point and radius is contained completely in the ellipse without touching it. If I could ...
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0answers
27 views

Reflective property of a hyperbola

An ellipse reflects an incident ray through one focus to the other as reflected ray and its special case of parabola likewise reflects rays parallel to symmetry axis after bouncing to go through its ...
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1answer
15 views

Coordinates of a string of beads

Given a number of circular beads of a given size strung on an ellipse with a known semi-minor axis, how can I calculate the position (let's say in Cartesian coordinates) of each bead's center? The ...
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1answer
24 views

Find one rational point on conics

We consider the equation: $Ax^2+Bxy+Cy^2+Dx+Ey-F=0$ with $A,B,C,D,E,F \in \mathbb{Q}$ If one has a rational point (a point whose coordinates are both rational) on the curve described by the ...
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2answers
25 views

Finding the major axis of an ellipse given the angle of a tangent

Given the angle between the tangent and the line that connects the point of tangency to each foci, and you are given the distance from one of the foci to the point of tangency. You are given two ...
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2answers
44 views

How to prove this theorem rhetorically?

It is not possible for a part of any of three conic sections to be an arc of a circle. It is asked to prove this theorem without using any notation or any modern form of symbolism whatsoever ...
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1answer
99 views

Volume of Revolution of a Transformed Ellipse?

I'm looking at an ellipse (a bunch of them actually) transformed by $h$ on the $x$-axis away from the center and rotated by an angle of $Q$ from the $xy$ axis. I got the following equation: the $x$ ...
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26 views

Tangents and Normals of rectangular hyperbolas

Please, could someone explain the solution to (d) I solved (a), (b) and (c) however, I don't understand how to calculate (d). (a) displayed in the question (b) $ q^2 y + x = 10q $ (c) displayed in ...
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1answer
26 views

Coordinate of $S(s,t)$ for Which Area of Quadrilateral is Maximum.

Let $P(-2,3)\;\;,Q(-1,1)\;\;,R()$ and $S(2,7)$ be $4$ points in order on the parabola $y=ax^2+bx+c$.Then the coordinate of $R(s,t)$ such that the area of Quadrilateral $PQRS$ has maximum ...
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1answer
16 views

Distance between feet of perpendiculars from focii of ellipse

$$Tangent\quad drawn\quad to\quad ellipse\quad { x }^{ 2 }+{ 2y }^{ 2 }=6\quad at\quad point\quad (2,1).If\quad A\\ and\quad B\quad are\quad the\quad feet\quad of\quad pependiculars\quad from\quad ...
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1answer
34 views

Points on a normal to a superellipse at distance $d$ from the curve

Given a point P0 $(x0, y0)$ lying on a super ellipse, $(x/a)^n + (y/b)^n = 1$, where $2 <= n <= 5$, I'm trying to derive an equation to describe the point P1 (x1, y1) lying on the normal through ...
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1answer
42 views

What's the importance of conics and quadrics in the context of a course of pure mathematics?

I've studied conics and quadrics in the past (specifically, in a course of analytic geometry). For these courses, we usually learn the basics of linear algebra and we apply these to conics and ...
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2answers
57 views

Find $a$, $b$ such that the ellipse $(x/a)^2 + (y/b)^2 = 1$ passes through $(\sqrt 2, 2)$ and has minimum area

I am working on a problem in which, for $a$, $b \gt 0$, we let $(x/a)^2 + (y/b)^2 = 1$ describe an ellipse. I am required to use the method of Lagrange multipliers and the corresponding second ...
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1answer
55 views

why there are no parabolic (on a paraboloid) non-euclidean geometry?

I have seen in many contexts that Euclidean geometry is called also "parabolic geometry". As in many things in mathematics (conics, differential equations, algebraic equations) the terms: ...
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27 views

Rewrite the equation of a conic in cartesian coordinates

Consider the equation for a conic in polar co-ordinates $(r,\theta)$ $$r = \frac{k}{1 - e\cos(\theta)} \qquad \qquad (1)$$ in the case where $k > 0$ and $e > 1$. Show that ...
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1answer
17 views

Find the width of an ellipse, on given Y position.

Look at picture below : Q: I wonder what's the equation to find $L$ (the red line on {x',y'}, which x' will always be $0$), ...
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4answers
387 views

How do I find the maximum perimeter of a rectangle inscribed in an ellipse?

The problem I've been stuck on is this: A rectangle is inscribed in the ellipse $$\frac{x^2}{20} + \frac{y^2}{12} = 1$$ What is the maximum perimeter of the rectangle? I don't even know if I'm ...
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1answer
21 views

When the equation of a conic becomes that of a pair of straight lines

This is a question I found in a book. Let $0<p<q$ and $a\neq0$ such that the equation $$px^2+4\lambda xy+qy^2+4a\left(x+y+1\right)=0$$ represents a pair of straight lines, then $a$ can lie ...
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0answers
33 views

On a non-standard approach to the classification of conics?

I've been introduced to a method of classifying conics but it's too cumbersome for me. I've discovered something that seems a little more promising on Eves' Elementary Matrix Theory: And ...
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1answer
41 views

Equation of Directrix of a Parabola

Find the equation of the directrix of the parabola $y^2+4y+4x+2=0$ I tried it as follows: $$(y+2)^2+4x-2=0$$ $$(y+2)^2=-4(x-\frac{1}{2})$$ On comparing with $Y^2=-4aX, a>0$, I got $a=1$ ...
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2answers
32 views

How to parameterise a parabola with a particle running on it at a given speed, say 10 m/s.

If I use t=x, it will dictate a uniform velocity in the x-direction, and eventually violate the constraint. Could use some hints. [edit: ... at a given speed, not velocity, my apologies].
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1answer
57 views

Fitting and Ellipse to a set of data points in Mathcad.

I would like to fit an ellipse to a set of data points in Mathcad and afterwards plot it. Searching the net, I stumbled on to Mike Shaw's post, which answers 75% of my question: See Plotting an ...
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2answers
56 views

Geometry: System of Circles

Given a circle $\,x^2+y^2+dx+ey+c=0,\,$ find the general equation of a circle passing through the intersection of this circle and the line $\,lx+my+n=0.$ My approach was to consider a circle of the ...
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2answers
32 views

Height of a rotated ellipse

If I have an ellipse, it is easy to find its height, twice the length of the major axis. But if the ellipse is rotated a certain number of degrees, how do you find the vertical height from top to ...
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1answer
75 views

Finding the length of semi major axis of an ellipse given foci, directrix and eccentricity

Can someone explain the solution to this. I don't get the assumptions in his solution. "A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is 1/2. Then the ...
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1answer
31 views

Foot of perpendicular on a chord of a conic

For a standard ellipse, a chord subtends an angle of $90^{\circ}$ with the centre $(0,0)$ . To find the locus of the foot of perpendicular to this chord from the centre of the ellipse, I wrote the ...
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52 views

Schwarz Function of an Ellipse

I want to find the Schwarz function of the ellipse define by $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b > 0. $$ To do so, substitute $$ x = \frac{z+\bar{z}}{2}, \quad y = \frac{z - ...
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1answer
42 views

Sketching the ellipse $|z-1| + |z-i| < 2 \sqrt{2}$

The problem asks to graph $|z-1| + |z-i| < 2 \sqrt{2}$. So I can tell this is an ellipse and that the distance from $(1,0)$ and $(0,1)$ sum to less than $2\sqrt{2}$. However, I can't see to get ...
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1answer
44 views

Conics - Locus of points

The ellipse with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ has one of its foci at the point $F$. The perpendicular from the origin to the tangent at a point $P(a\cos\theta, b\sin\theta)$ on the ...
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1answer
33 views

For all unit vectors $\mathbf u$ and a positive definite $\mathbf C$, what surface do vectors $\mathbf u \mathbf u^\top \mathbf C \mathbf u$ form?

Let $\mathbf C$ be a positive-definite $k\times k$ matrix. For all vectors $\mathbf u\in \mathbb R^k$ of length $\|\mathbf u\|=1$, consider vectors $\mathbf {uu}^\top\mathbf{Cu}$; they form a surface ...
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2answers
17 views

Finding the focus points of a hyperbola

So I have the following hyperbola : $\frac{x^{'2}}{4}-\frac{y^{'2}}{4}=-1$ I need to find the focus points of this hyperbola. What is some analytical way to do this ? Thank yoU!
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2answers
19 views

Parametric Equation of conics: Parabola

Let $P(ap^2,2ap)$ and $Q(aq^2,2aq)$ be two points on the parabola $y^2=4ax$ such that PQ is the focal chord. Let $A(at^2,2at)$ and $B(as^2,2as)$ be two other variable points on $y^2=4ax$. a) Show ...
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3answers
62 views

Finding the vertex and focus of a rotated parabola

So I begun with the following equation : $x^2+2xy+y^2+2\sqrt{2}x-2\sqrt{2}y+4=0$ I transformed it in the following : $y'=\frac{x'^2}{2}+1$ I had to do a rotation of $\frac{\pi}{4}$ of the xy axis. ...
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2answers
36 views

Eliminating the $xy$ term of an equation for a conic gives $\tan 2\beta = 2/0$

So, I have the following : $$x^2 + 2xy + y^2 + 2\sqrt{2}\;x - 2\sqrt{2}\;y+4=0$$ I know that to take out the $xy$ term from $A x^2 + B x y + C y^2 + \cdots = 0$, I must use : $$\tan ...
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2answers
120 views

Eccentricity of an ellipse.with b > a

I have the following ellipse : $\frac{(x-3)^2}{\frac{9}{4}} + \frac{(y+4)^2}{\frac{25}{4}}=1$ In this case, b > a. It says that to find the eccentricity I must use $\frac{c}{a}$ but I think this is ...
2
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0answers
25 views

Coordinates of a plane with a handle

I am trying to find the appropriate coordinates for a plane with a handle (of topology $\mathbb{R}^2 \# \mathbb{T}^2$), without having to use several coordinate patches. My current intuition is to ...
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1answer
20 views

conic with locus equation

Find the equation of the locus of the point P(x, y) such that the sum of its distance to the points A(6,0) and B(-6, 0) is 18 units. answer ((x-6)^2+y^2)^1/2 + ((x+6)^2+y^2)^1/2 = 18 ...
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0answers
10 views

Keeping a parabola's roots after vertical shift

Suppose f(x) = -x(x - a) + b, where a > 0 and b >= 0. When ...
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1answer
37 views

Equation relating square of distance to point and distance to line

Find the equation of the locus of the point $P(x, y)$ such that the square of the distance from $(-2, -5)$ to $P(x, y)$ is three times the distance from $P(x, y)$ to the line $8x+15y=34$. My answer: ...