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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Hyperbola application

A curved mirror is placed in a store for a wide angle view of the room. the right hand branch of x squared over one minus y squared over three equals one models the curvature of the mirror. a small ...
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Find the area of $4x^2-2xy+y^2=1$ [on hold]

Any help? Ive tried everything I can think of.
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56 views

Understanding Conics in Pencil

In a paper I'm reading about ellipses they talk a lot about "pencils of conics", after looking around on the web to learn more like this website: http://planetmath.org/pencilofconics I found some ...
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Why is $ \theta(m) \propto \zeta(2) $ if it is counting lattice points in a hyperbola?

I found this lattice point identity in a derivation of $\zeta(2)$: $$ \theta(x) = \sum_{mr \leq x} m = \sum_{r \leq x}\sum_{m=1}^{[x/r]} m = \sum_{r \leq x} \left( [x/r]^2 + [x/r] \right) = \sum_{...
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1answer
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If tangents are drawn from two points which are equidistant from given point, then find the locus

Tangents are drawn to the circle $x^2+y^2=a^2$ from two points on the $X$ axis equidistant from the point $(k,0)$ prove locus of their intersection is $ky​^2=a^2(k-x)$. If I take points as $(k+\alpha,...
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Hypocycloid with an outer ellipse

I have tried to change the traditional hypocycloid a bit. What I've basically done is that a circle now rolls inside an ellipse. I am trying to find the equation for the same. I am mostly done, ...
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Predict the range of a ball kicked on the moon

I'm very confused over this question I had in my homework (someone told me you can't ask hw questions but I really want to know how to do this and I don't have access to a teacher right now) Sorry if ...
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Football on the moon [on hold]

I'm very confused over this question I had in my homework (someone told me you can't ask hw questions but I really want to know how to do this and I don't have access to a teacher right now) Suppose ...
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2answers
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determine all (x,y) of the line Normal to an Ellipse

Hi everyone I have a question that requires me to determine the (x,y) coordinates of all points that intersects the x-axis on this ellipse when the normal line has a slope of -4, and I'm curious to ...
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1answer
22 views

Why Circle is traced counterclockwise and ellipse is traced clock wise?

In the Lecture 32: Polar Coordinates,professor traces the circle counterclockwise, but traces the ellipse clockwise. "Which was this one here. And first we noted that this does parameterize, as we ...
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84 views

circle tangent to three circles

To-day I want to look at CCC - one circle tangent to three circles whose radii and positions of their centers are known. How does one solve this.. old fashioned ways like ruler and compass, or ...
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1answer
16 views

Calculate the Y value of an arc of an ellipse given X

Before I say anything else, let me just state, that I am only in high school and going in Precalculus this upcoming school year, which is probably why I am having this issue in the first place. And ...
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How to calculate $\Delta$ in conic sections?

When learning conic section I learnt about $\Delta$. For any conic in general form : $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ Here $\Delta=abc +2fgh - af^2 - bg^2 -ch^2$ The conic is said to be ...
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1answer
54 views

Determine the largest area of an ellipse enclosed by the hyperbolas ($xy=1$ and $xy=-1$)

Question: An elipse with equation $$ {x^2\over a^2} + {y^2\over b^2} = 1 $$ is enclosed by the hyperbolas given by $xy=1$ and $xy=-1$. , Determine the largest area of an ellipse enclosed by the ...
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3answers
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Why is (h,k) in the Vertex Formula of a Parabola y=a(x-h)^2 +k the vertex?

Why is it that in the form y=a(x-h)^2+k of a parabola, that (h,k) is the coordinate of the vertex? I am reviewing Algebra and cannot find a reasonable explanation anywhere.The highest level of math I ...
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1answer
30 views

Parametric equations of an ellipse in an arbitrary plane at an arbitrary orientation?

I have searched both on here and on stackoverflow for answers to this question and I can't seem to find a good answer relating to what I'm doing. I have a center point and two vectors, one that is in ...
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101 views

Conjecture about circles in plane lattices

A plane lattice $\Lambda$ is a set $\Lambda= \{ mA+nB: m,n \in \mathbb Z \}$, where $A,B$ are linearly independent vectors in $\mathbb R^2$. The set of all circles in $\Lambda$ is $$\mathcal K(\...
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3answers
35 views

What is the equation of this hyperbola?

What is the equation of the hyperbola that satisfies these conditions: Asymptotes $y=2x$ and $y=-2x$, centre $(0,0)$, and the point $(1,1)$ lies on the curve. This isn't a homework question; I study ...
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1answer
48 views

Find point on ellipse arc

Known ellipse semiaxis, points F and G, angles alpha and beta. How to find point H?
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1answer
38 views

Geometric proof re: parabola focus and potential calculus connection

The image above is a proof that light traveling with an orientation perpendicular to the directrix into a parabola will be reflected to the focus of the parabola. (In the diagram, the purple dotted ...
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2answers
45 views

translation and rotation of a parabola

I am trying to translate a parabola to the origin, rotate by T radians and then translate back to the original position. I can calculate the new X and Y vectors using matrix operations and the regress ...
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1answer
69 views

Get all points in an ellipse knowing the center, one point, the vertical axis and the horizontal axis

How to get all the points in an ellipse when I know a point of it and it's center? I have the following situation: enter image description here I know the position of the red dot relative to the ...
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1answer
32 views

Intuitive explanation of Pascal's Theorem

I am wondering why Pascal's Theorem, as well as other 'Euclidean' results in projective geometry like Brianchon's Theorem should be true for not only circles, but also conics in general. Is there ...
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1answer
46 views

Finding the parameters of an ellipsoid given its quadratic form

Suppose we have the quadratic form of an ellipsoid of the form $$ax^2 + by^2+cz^2+dxy+eyz+fxz+gx+hy+iz+j=0$$ I want to find centroid of the arbitrarily oriented ellipsoid, its semi-axes, and the ...
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1answer
61 views

Is ellipse intersecting with circle?

I have circle given by center coordinates and radius, and ellipse with center coordinates, $r_x$ and $r_y$. I want to check if the ellipse is inside the circle( meaning their bounds can collide). How ...
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1answer
62 views

Ellipse and parallel lines

Let's imagine that we have an ellipse described by the known equation $v^TAv=0$, (Link_1) where $v=[x \ y \ 1]^T$ (it can be a skew one in a general case). Then we have all possible parallel lines - ...
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Points at infinity of a conic section and its eccentricity, foci, and directrix?

Background on projective geometry and conic sections; you might want to skip to the actual question A conic section is analytically described as the zero-locus of points $(x,y)$ in the affine plane ...
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1answer
16 views

Eliminate asymptote using projective transform

I have a well-behaved curve $f:\mathbb{R}\rightarrow \mathbb{R}^2$ which has exactly one linear asymptote passing through points $p$ and $q$ in $\mathbb{R}^2$. I would like to find a projective ...
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3answers
83 views

Show that the circle drawn on a focal chord of a parabola $y^2=4ax$, as a diameter touches the directrix

Question: Show that the circle drawn on a focal chord of a parabola $y^2=4ax$, as a diameter touches the directrix. Let the parabola be $y^2=4ax$ Let the focal chord be $y = m(x-a) $ Subbing ...
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Calculate Ellipse From 5 Points

How can I find a general or parametric form of equation for the ellipse having 5 points that lie within that ellipse? I have found this solution: Calculate Ellipse From Points?, where unfortunately ...
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3answers
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How to get the correct angle of the ellipse after approximation

I need to get the correct angle of rotation of the ellipses. These ellipses are examples. I have a canonical coefficients of the equation of the five points. $$Ax ^ 2 + Bxy + Cy ^ 2 + Dx + Ey + F = 0$...
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1answer
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Question about vectors, planes and lines

Let $O,A$ be two points in the plane with $|\vec{OA}|=3$. Which line is formed by the points $M$ of the plane, such that $\vec{OM}(\vec{OM}-2\vec{OA})=7$ ? My attempt.. Suppose $\vec{OM}=\vec{x}$ ...
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1answer
58 views

Integer Solutions to an Ellipse

I'm trying to find positive integer solutions to the ellipse $$x^2 - xy + y^2 - k^2 = 0$$ where $k$ is a constant. Specifically, I already have two solutions for a given $k$, and I'm trying to find a ...
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Applying drag to a collision prediction formula

I feel like this question might be below the minds of Math StackExchange, but I'll try anyway. (I can understand Math generally, but I'm probably not the caliber of people here.) I've been working on ...
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2answers
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Prove that the directrix-focus and focus-focus definitions are equivalent

(NOTE: This is my attempt at answering this question and this question, but I rewrote it in order to make it easier for me to solve. Also, I've made a YouTube video explaining this whole proof. Note ...
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Poncelet's closure theorem

Need some help understanding the proof made by Kneebone and Semple in "Algebraic Projective Geometry". I loose it in the sentence about the (2,2) correspondance. As I understand it, they setup an ...
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Find the plane which touches the cone $x^2+2y^2-3z^2+2yz-5zx+3xy=0$ along the generator whose direction ratios are $1,1,1.$

Find the plane which touches the cone $x^2+2y^2-3z^2+2yz-5zx+3xy=0$ along the generator whose direction ratios are $1,1,1.$ Let the plane touches the cone at $(\alpha,\beta,\gamma)$. We know that ...
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1answer
59 views

Given the bounds of a rotated ellipse, can you find the semi-major and semi-minor axis?

Clarification: I am trying to find the semi-axes $(a,b)$ given the bounding rectangle's dimensions $(x,y)$. To constrain the problem, I am keeping $\theta$ the same as my original ellipse. The ...
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1answer
70 views

Discriminant of a Conic Section

$B^2 - 4AC$ is called the discriminant of a conic section. It is an invariant. Depending on the sign of $B^2 - 4AC$, you can tell which of the three conic sections (Ellipse, Hyperbola, Parabola) where ...
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Find the length of the chord joining the points in which the straight line $\frac{x}{a} + \frac{y}{b}= 1$ meets the circle $x^2+y^2=r^2$

Question: Find the length of the chord joining the points in which the straight line $\frac{x}{a} + \frac{y}{b}= 1$ meets the circle $x^2+y^2=r^2$ My initial thoughts were rearranging the ...
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1answer
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Planes through $OX$ and $OY$ include an angle $\alpha,$ show that their line of intersection lies on the cone $z^2(x^2+y^2+z^2)=x^2y^2\tan^2\alpha$

Planes through $OX$ and $OY$ include an angle $\alpha,$ show that their line of intersection lies on the cone $z^2(x^2+y^2+z^2)=x^2y^2\tan^2\alpha$ The lines of intersection of the planes through $...
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79 views

Show that the vertex lies on the surface $z^2(\frac{x}{a}+\frac{y}{b})=4(x^2+y^2)$

Two cones with a common vertex pass through the curves $z^2=4ax,y=0$ and $z^2=4by,x=0.$ The plane $z=0$ meets them in two conics which intersect in four concyclic points.Show that the vertex lies on ...
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2answers
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Prove that the equation of the cone $yz(\frac{b}{c})+zx(\frac{c}{a}+\frac{a}{c})+xy(\frac{a}{b}+\frac{b}{a})=0$

The plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ cuts the coordinate axes in $A,B,C.$Prove that the lines passing through the origin and intersecting the circle $ABC$ generate the cone $yz(\frac{b}{c}...
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quadratic function vs conic section

I am categorizing types of math problems on the ACT. I started off with 'quadratic function' as one category, and 'conic sections' as another... It seemed like a simple classification at first, but ...
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How do you find 1 standard deviation from the center of an ellipse along the 45-degree angle?

I have generated a normally-distributed elliptical cluster of data in Matlab. The center of the ellipse falls at (0,0) and it has a standard deviation of 1 along one principle axis and a standard ...
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2answers
102 views

How to get the coordinates of the center of the ellipse after approximation

I create an algorithm recognizing ellipses in images. I have five coordinates (points) possible ellipse. (8.8) (7.4) (6.3) (3.6) and (2.2) I use the formula of the conical section of the ...
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66 views

Finding centers of ellipses with two points and their respective tangents

I hope you can help me with the following, probably rather complex dilemma: I generally want to find an ellipse given two points and their respective tangents in 2-D space (X and Y coordinates). Now ...
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4answers
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Common tangents to circle $x^2+y^2=\frac{1}{2}$ and parabola $y^2=4x$

I'm having trouble with this. What i do is say $\epsilon: y=mx+b$ is the tangent and it meets the circle at $M_1(x_1,y_1)$, i equate the $y$ of the tangent with the circle: $y=\pm \sqrt{1/2-x^2}$ and ...
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1answer
21 views

Spaces relations

I have a physics question for which I need to determine the radius of a circle. Given are two Ellipse shapes with the same center (0,0 in a Cartesian space). the Height and Width of the smaller is 1[...
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How do I find the left end height of an ellipse tilted at the right end?

Let's say I have a large ellipse on a flat ground as shown in Figure 1 and 2 below. Angles DBC and ABD are right-angles and the longest radius with respect to the ground, BD is perpendicular to line ...