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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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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2answers
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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 ...
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1answer
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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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1answer
499 views

Volume of ellipsoid bounded by two planes.

I need to find the volume of ellipsoid: $$5x^2 + {y^2\over25} + {3z^2\over4} = 1$$ if the ellipsoid is bounded by $z={-1\over2}$ and $z=1$ planes. I was able to find the total volume of the ellipsoid ...
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4answers
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Does a right circular cone only consists of pair of straight lines, hyperbolas, parabola, circles and ellipses? [closed]

I was reading about the conic sections and that a conic section includes pair of straight lines, ellipses, hyperbola, circles and parabola. Are all these 5 components enough to form a right circular ...
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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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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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81 views
+100

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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714 views

Definition of an ellipsoid based on its focal points

I have a question concerning the formulation of an (3D) ellipsoid. The most common definition for an ellipsoid seems to be: $E = \{ x=\left( x_1, \dots x_n \right)^T \in R^n: \sum_{i=1}^n \left( \...
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1answer
56 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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2answers
520 views

Is it possible to calculate the volume of a parabolic arch?

Given that you know the equation of a parabola that only has positive values, is it possible to find the volume of the parabolic arch itself? NOT the volume of space underneath the arch. I asking ...
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1answer
24 views

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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Finding Asymptotes of Hyperbolas

To find a asymptote its either b2/a2 or a2/b2 depending on the way the equation is written. With the problem $$\frac{(x+1)^2}{16} - \frac{(y-2)^2}{9} = 1$$ The solutions the sheet I have is giving ...
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1answer
54 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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1answer
538 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 ...
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4answers
246 views

How to find the area enclosed by the ellipse $b^2x^2 + a^2y^2=a^2b^2$?

I need to find the area enclosed by the ellipse $b^2x^2 + a^2y^2=a^2b^2$, and I know it involves taking the integral, but I'm not sure what function I should be taking the integral of or how to find ...
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1answer
2k views

How to compute the chord length of an ellipse?

How do I calculate the chord length of ellipse? I need to design a blade vane rotating inside an elliptical profile. The blade is supposed to be in contact with ellipse with a maximum clearance of $0....
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1answer
57 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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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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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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1answer
75 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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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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1answer
681 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
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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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4answers
129 views

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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4answers
81 views

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
60 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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1answer
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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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1answer
515 views

Computing a matrix to convert an (x,y) point on an ellipse to a circle

I have an ellipse defined by its semi-major axis, inclination, and position angle. The ellipse is centered on the origin. I would like to solve for a matrix that converts this ellipse to a circle. ...
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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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1answer
512 views

Equation of normal vector pointing away from ellipse

Assuming that I have an ellipse, centered at $(h,k)$ of type: $$\left(\frac {x-h}{a}\right)^2 + \left(\frac {y-k}{b}\right)^2 = 1$$ The gradient of the normal is: $$\frac{a^2(y-k)}{b^2(x-h)}$$ ...
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1answer
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Radius vs Radius of curvature of an ellipse

I am a bit confused by the physical meaning of radius vs radius of curvature, with regards to an ellipse. For a standard ellipse: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ In this case, the $a$ ...
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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
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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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2answers
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3-D Geometry Problem. Find a curve which touches the straight line.

If two perpendicular tangent planes to paraboloid $x^{2}+y^{2}=2z$ internsects in a straight line in the plane $x=0$, obtain the curve to which the straight line touches. I don't know how to ...
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1answer
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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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Slope of common chord of 2 parabola one having $x$-axis as directrix another having $y$-axis as directrix

Two parabolas $x^2=4ax$ and $y^2=4ay$ have common focus and have $x$-axis and $y$-axis as their directrix respectively the slope of their common chord is ?
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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 ...
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4answers
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How to find the equation of a parabola when only given the x-intercepts and the Axis of symmetry?

i have been given the following problem. A tennis ball is lobbed from ground level and must cover a horizontal distance of 22m if it is to land just inside the opposite end of the court. If ...
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How to find the center of an ellipse?

I have the following data:- I have two points ($P_1$, $P_2$) that lie somewhere on the ellipse's circumference. I know the angle ($\alpha$) that the major-axis subtends on x-axis. I have both the ...
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1answer
30 views

Points on the curve

We have to find points on the curve $ax^2+ay^2+2 bxy=c$ (Where c>b>a ) whose distance from origin is minimum . I am not getting any start . I am able to just find that the curve would be hyperbola
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Equation of common tangent.

What is the equation of common tangent to the circle $(x-3)^2+y^2=9$ and parabola $y^2=4x$.$$My Try$$ So equation of tangent at point $(x_1,y_1)$ is $xx_1+yy_1-3(x+x_1)=0,yy_1=2(x+x_1)$ for circle,...
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Possible generalization of decimal expansion of $\frac{1}{7}$ on an ellipse

My question: Is there a generalization of the result below, either involving more digits or a fraction other than $\cfrac{1}{7}$? The Futility Closet has this surprising result: The One-Seventh ...
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How to think of 2 intersecting planes in $\mathbb{R}^3$ as a cone?

It is well known that any (possibly degenerated) conic section in $\mathbb{R}P^2$ is given by, up to a projective transformation, a point, a line, two lines or a circle (given by the equation $x_0^2+...
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1answer
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Length of normal. chord…

What is the length of normal chord which subtends right angle at the vertex of parabola $y^2=4x$. $$My Try$$ let the equation of normal be $y=mx-am^3-2am$ Now I assumed slope of this line as $45$ (...
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How to find an ellipse , given 2 passing points and the tangents at them?

Please answer to a question , how to find an ellipse which passes the 2 given points and has the given tangents at them. And one related question is that the given condition can decide just one ...
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Can I define a 2-D ellipse via two points and their respective tangents? [duplicate]

I have encountered a rather complex issue that I could not solve on my own, and my research online has so far not yielded a suitable answer to my question, hence my decision to seek help in this place....
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1answer
142 views

Equation for the length of a chord parallel to either the minor or major axis in an ellipse

I am looking for a way to compute the length of any chord parallel to the minor (or major) axis of an ellipse. In all cases I know the lengths of both axes, and the distance between the chord and axis ...
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33 views

What is the shape of an ellipse (or parabola) that has rotated around the x-axis?

What is the shape of an ellipse (or parabola) that has rotated around the x-axis? Is there a specific name?