# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...