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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1answer
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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
23 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 $\vec{OM}=\vec{x}$ ...
2
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1answer
55 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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0answers
54 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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2answers
51 views

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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0answers
25 views

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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0answers
23 views

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
56 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
52 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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4answers
80 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
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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
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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2answers
62 views

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
36 views

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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0answers
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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
82 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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1answer
59 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
127 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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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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1answer
20 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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1answer
31 views

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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2answers
122 views

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
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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2answers
73 views

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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0answers
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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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1answer
37 views

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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0answers
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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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4answers
57 views

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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0answers
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?
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1answer
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Is there any equation for this type of skewed parabola?

I am having the following parabola looking curve (blue curve), but its not exactly symmetric. The best fit that I am getting using a quadratic equation is also shown (black curve) but it is not ...
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1answer
34 views

projection of an ellipsoid on XY plane

The equation of an ellipsoid is $$ax^2+by^2+cz^2+2fyz+2gxz+2hxy+2px+2qy+2rz+d=0$$ The ellipsoid is arbitrary rotated and the orientation angle are given as θ, β and Ѱ and the center is at (x',y',z')....
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0answers
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equation of parabola if parametric coordinates are given

The length of focal chord of the parabola given by $x=pu^2+qu+r$ and $y=p'u^2+q'u+r'$ Where $u$ is a variable $\bf{My\; Try::}$ Given $pu^2+qu+(r-x)=0$ and $p'u^2+q'u+(r'-x)=0$ So Using Cross ...
7
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1answer
137 views

Photo image to find the screen orientation

I am trying to find the angle of tilts of a screen using projection of a circle from a source $S$. The light beam falls on the photo screen to expose it and what we get is an ellipse with major axis $...
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1answer
52 views

Maximium and minimum value of area.

Given that the equation of parabola is $y=x^2+1,1\leq x\leq 3$ What is the maximum and minimum value of area formed by x-axis,tangent,normal at any point on parabola. Now I wrote the equation as $x^2=...
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4answers
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For the general ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ,show that the midpoints of the chords lie on a straight line.

Question: A collection of parallel chords connect pairs of points on an ellipse, as shown For the general ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, ...
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29 views

Value of the integral in the interval.

The length of perpendicular from foci $S,S'$,on any tangent to ellipse $$\frac{x^2}{4}+\frac{y^2}{9}=1$$ are $a,c$ respectively then value of $\int \{x\}dx$ from $-ac$ to $ac$ is ?(where {} denotes ...
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1answer
51 views

General Conic and its Rational Solutions

Suppose you have a rational conic $ax^2+bxy+cy^2+dx+ey+f=0$. There is a theorem that states if a conic has 1 rational solution it has infinitely many rational solutions. How can you prove this ...
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Equation of tangent line to an ellipse, having a given angular coefficient

I've got to find the equations of the tangent lines to the ellipse: ${ x^2 / a^2 + y^2 / b^2 - 1 = 0 }$ having a given angular coefficient m that is a real number. I search for angular coefficient,...
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1answer
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Derive the length of the longest line segment that can be enclosed inside the region A.

Q. Let A be the region in the xy-plane given by A={(x,y): x=u+v, y=v, u^2+v^2≤1}. Derive the length of the longest line segment that can be enclosed inside region A. My attempt: I found the equation ...
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0answers
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Martini Glass - Extension

This is my extension to the very interesting question on the martini glass from 538.com by the Riddler as posted here earlier by MP Droid. Recap of original configuration. A martini glass has the ...
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0answers
33 views

Proof of the Discriminant Law of Conics

Does anyone know a good resource and or know the proof for the Discriminant Law of Conics ($B^2 - 4AC > 0$ , Hyperbola ....) Thanks
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1answer
47 views

Conics and conics of the form $ax^2+by^2+c=0$

The problem of finding rational points on conics is usually discussed (for example in the book of Silverman and Tate) for conics of the form $ax^2+by^2+c=0$. I assume that those conics are in ...
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4answers
94 views

Find the minimum distance to move an ellipse to be inside another ellipse?

For the problem of ellipse intersection, I would like to know an accurate "general, including the cases of two non intersected ellipses, and non aligned ellipses" method to calculate the minimum ...
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1answer
21 views

Point of intersection of ellipses

If two ellipses are intersecting at a point,is it necessary that the line drawn joining the centre of those two ellipses should also pass through the point of intersection (of ellipse)? (if yes,how to ...
1
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1answer
24 views

Calculator limits on a parabola

Hi guys I'm making Patrick Star for a graphing project. Anyways I'm using a parabola for his head on my TI-84 but when I set limits on it, it graphs a straight line. So the equation itself is $y=(-0....
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2answers
108 views

Minimum Area of An Ellipse Surrounding Four Circles

The circles are all four combinations of $(x\pm60)^2+(y\pm25)^2=5^2$ (see pic at end). The ellipse I've got is one I found via trial and error but there must be an analytical way to solve this, right?...
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3answers
76 views

How to show that any rectangle in ellipse must be oriented parallel to axes?

A problem which is often given as an exercise for students learning about calculus and finding extrema, is to find maximal possible area of a rectangle inside an ellipse. Such question was asked, for ...
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2answers
46 views

Conic Sections: Hyperbola (Finding the Locus)

This is a multipart question so bear with me until I get to the part where I am stuck on. $H$: $xy=c^2$ is a hyperbola (i) Show that $H$ can be represented by the parametric ...
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0answers
28 views

Rotate an Ellipse

$x = h + a \cos(φ) \cos(θ) + b \sin(φ) \sin(θ)$ $y = k + b \cos(φ) \sin(θ) - a \sin(φ) \cos(θ)$ Hi, I have basic question of parametric equation for ellipse. I'm trying to rotate horizontal ellipse ...