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Questions tagged [conic-sections]

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.

3
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22 views

Smallest ellipse by area to enclose 2D points

I am interested in writing an algorithm that, for a given set of $n$ points in $\mathbb{R}^2$, finds the equation of the smallest ellipse by area to enclose all $n$ points. Here's where I'm at: The ...
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votes
2answers
31 views

Determine the coefficient a,b,c

Make the parabola $y=a x^2+b x+c$ pass through $(2,1)$ and the tangent to the line $y=2x+4$ at $(1,6)$. Please explain it step by step.
2
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1answer
51 views

A concave quadrilateral cannot be circumscribed by an ellipse nor by a parabola.

It's intuitive that a concave quadrilateral cannot be circumscribed by an ellipse nor by a parabola. Is there a formal proof of it?
2
votes
2answers
59 views

The cyclic quadrilateral and the slopes of its sides

Suppose a plane quadrilateral ABCD (convex, concave or crossed) no side of which is parallel to y-axis, and let $m_1, m_2, m_3, m_4$ be the slopes of the equations of sides AB, BC, CD, DA. Having ...
2
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2answers
41 views

If $4\alpha^2–5\beta^2+6\alpha+1=0$.Prove that $x\alpha+y\beta+1=0$touches a Definite circle. Find the centre and radius of the circle.

If $4\alpha^2–5\beta^2+6\alpha+1=0$. Prove that $x\alpha+y\beta+1=0$touches a Definite circle. Find the centre and radius of the circle. I tried to solve this question by taking a General equation of ...
0
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1answer
17 views

Vertical cut-off distance to transform vertical parabola into rotated circle segment

Given a known vertical, negative parabola ($y=ax^2+bx+c$ with $a$,$b$ & $c$ known parameters): how far down from the vertex of the parabola (the top) is the vertical distance, so when you "cut if ...
1
vote
1answer
51 views

An interesting (conjectural) property of any triangle

Given any triangle $\triangle ABC$, we can always build three ellipses, each of them having foci in two of the vertices and passing through the third one, as shown in the following picture: In ...
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0answers
26 views

I got 5 ellipse parameters, 4 of which are coordinates of 2 focus points, what would the last one be? And how can I get the ellipse function?

I am working on a machine learning project processing fundus images. I have an image dataset, each image in which comes with a 5-number tuple called "ellipse parameters". However, the dataset is ...
0
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1answer
32 views

Parallel surface to paraboloid

I have the equation for a paraboloid $z=x^2+y^2$ I need to obtain a parallel surface, Does anyone know what is the procedure to determine it?
2
votes
1answer
49 views

A conjecture involving three parabolas intrinsically bound to any triangle

Given any triangle $\triangle ABC$, we can build the parabola with directrix passing through the side $AB$ and focus in $C$. This curve intersects the other two sides in the points $D$ and $E$. ...
2
votes
1answer
71 views

An ellipse intrinsically bound to any triangle

Given any triangle $\triangle ABC$, we build the hyperbole with foci in $A$ and $B$ and passing through $C$. The hyperbole always intersects the side of the triangle that is opposite to the vertex ...
9
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2answers
131 views

A conjecture about the intersections of three hyperboles related to any triangle

Given any triangle $\triangle ABC$, we build the hyperbole with foci in $A$ and $B$ and passing through $C$. Similarly, we can build other two hyperboles, one with foci in $A$ and $C$ and passing ...
0
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1answer
36 views

Inscribed triangle in an ellipse

An ellipse has eccentricity $0.8$ and a line bisecting the semi-minor axis $AB$ perpendicularly cutting the ellipse at point $O$. Find angle $\hat{AOB}$.
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1answer
17 views

Why does $b^2+a^2m^2=c^2$ in coordinate systems?

I'm wondering how the formula above is derived. This is when the equation of an ellipse is $$(x^2/a^2)+(y^2/b^2)=1$$ and the tangent has an equation of $$y=mx+c$$
3
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3answers
84 views

Parameterizing lines reflected in a parabola

Points reflected by a parabolic mirror create images that appear to be at specific positions on the other side of the mirror. I am attempting to use geometry to find the map from each point in space ...
1
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4answers
87 views

Condition on $k$ for $x^2+y^2-12x-6y-4=0$ and $x^2+y^2-4x-12y-k=0$ to have simultaneous solutions $(x,y)$

The two equations: $$x^2+y^2-12x-6y-4=0$$ and $$x^2+y^2-4x-12y-k=0$$ have simultaneous real solutions $(x,y) \iff a \le k\le b$. Then, what is the value of $a+b$?
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0answers
66 views
+50

Locus of boundary when shadow is taken

For the first part (i), I could solve by taking images and I got the answer as ellipse, but for the second part I don't know how to take shadow. Can I get exact equation of locus or do I get just ...
0
votes
1answer
23 views

How can I create a quadratic equation that flips the parabola, starting at a certain arbitrary point near the vertex?

Given an arbitrary quadratic polynomial $$ax^2+bx+c$$and a corresponding graph where gray lines are axis, black is the parabola, red is an arbitrary line near the vertex, How would you go about ...
0
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0answers
19 views

Condition for a line to be tangent to a parabola and normal to another one simultaneously

Question Find the condition for a line other than y-axis to exist such that it is tangent to $y^2=4ax$ and normal to $x^2=4by$. Attempt For tangent line: $$x-y{t_1}+a{t_1}^2=0$$ where $t_1$ is ...
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1answer
101 views

If the equation $\alpha x^2+4\gamma xy+\beta y^2+4p(x+y+1)=0$ represents a pair of lines. Find the range of $p$ in terms of $\alpha,\beta$

For $\alpha,\beta,\gamma\in\mathbb{R}$ with $0<\alpha<\beta$, if $$\alpha x^2+4\gamma xy+\beta y^2+4p(x+y+1)=0$$ represent a pair of lines. Then which one is right? (a) $p\in[\alpha,\...
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0answers
19 views

Tangent to an ellipse

My question is if the tangent and normal to an ellipse meet the x axis at T and N respectively. How do I show that OT*ON is a constant? Whereby O being the origin.
2
votes
1answer
38 views

How do you substitute an arbitrary conic equation with trigonometric functions?

For example, Circle ($\frac{x^2}{a^2}+\frac{y^2}{a^2}=1$) : $$ \left\{ \begin{array}{c} x\to a\cos \theta \\ y\to a \sin \theta \\ \end{array} \right. $$ Oval ($\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$) : ...
3
votes
2answers
62 views

Conics based on Intercepts

$\hspace {2cm}$ We know that the ellipse $$\frac {x^2}{a^2}+\frac {y^2}{b^2}=1\tag {1}$$ intercepts the axes at $(\pm a, \pm b)$. It is interesting to note that the parabola $$\frac {x^2}{a^2}+\...
0
votes
0answers
41 views

How, to interpret the geometry of a 2nd degree polynomial equation, which contains an elliptic quadratic form and whose constant coefficient is zero?

$$ \mathbf x = \begin{bmatrix} x & y \end{bmatrix}^\mathsf T $$ $$ Q = \begin{bmatrix} A & B/2 \\ B/2 & C \end{bmatrix} $$ Eigendecomposition of $Q$: $ ~Q = \begin{...
0
votes
1answer
33 views

Find the equation of the parabola in which the ends of the latus rectum

Find the equation of the parabola in which the ends of the latus rectum have the coordinates $(-1,5)$ and $(-1,-11)$ and the vertex is $(-5,-3)$. I could think of assuming the equation of parabola as ...
0
votes
0answers
17 views

Finding Vector3 of a rotated ellipse in 3D space

I have the following code to evaluate an (X, Z) point on an Ellipse given a t-value between 0 and 1. I want to be able to rotate the ellipse by a certain number of degrees along the minor axis. <...
0
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0answers
27 views

What is the focal length of an arbitrary parabola? [duplicate]

Given a parabola of the form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, where $B^2 - 4AC = 0$, what is the formula for the focal length?
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2answers
36 views

Calculate angle between a chord and the circle's origin

I need to make an arc that has a varied size. For that I need to calculate the starting and ending angle of the chord cutting the arc. How can I do that knowing only the radius and the chord's length. ...
1
vote
1answer
28 views

A Poncelet-style theorem about tangents from a point to an ellipse - is it known?

I just proved, using lengthy and awkward trigonometric calculations, the following theorem: Let $\mathcal{E}$ be an ellipse with foci $E$ and $F$, let $P$ be a point outside the ellipse, and let $...
2
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0answers
33 views

How many foci are there in conics?

As per book every conic have 4 foci ,two real and 2 imaginary. I cannot understand and visualize this.
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2answers
42 views

If the tangents to parabola $y^2=4ax$ at $(at^2,2at)$ and $(as^2,2as)$ meet at $(p,q)$, then $a^2(t-s)^2=q^2-4ap$

If the tangents to the parabola $y^2 = 4ax$ at the points $(at^2, 2at)$ and $(as^2, 2as)$ meet at the point $(p, q)$. Show that $$a^2(t - s)^2 = q^2 - 4ap$$ My work so far: Using $yy = 2a(x + x)$ ...
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1answer
51 views

Distance of normal to the ellipse from the centre of the circle

Find the normals to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ which are farthest from the centre. My approach Let the point be $(x_1 ,y_1)$ and the tangent equation is $\frac{xx_1}{9}+\frac{yy_1}{...
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vote
1answer
35 views

Proving non-centered at the origin ellipsoid A is a subset of ellipsoid B

I would like to prove an ellipsoid $E_a$ center at $q_a$ with shape matrix $A$ and radius $r_a$ defined as: $(x-q_a)^T A^{-1} (x-qa) <= r_a$ is included into another ellipsoid $E_b$ -i.e. $E_a$ ...
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0answers
16 views

name for a curve ($1/(1+x^n)$) related to the rectangular hyperbola

This question concerns the nomenclature of curves related to hyperbolae. The expression $\frac{A}{x-c_0}+d_0$ corresponds to a rectangular hyperbola, which for the choice of parameters $A=1, c_0=-1, ...
1
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1answer
66 views

Projectile Envelope

Consider a projectile launched from the origin at velocity $v$ and angle $\theta$. From this other question we see several approaches to arrive at the equation for the envelope of different ...
3
votes
1answer
64 views

Solve $a^2 + 2b^2 - 3c^2 - 6d^2 = 1$ and $ab=3cd$ over integers

I'd like to solve this set of equations over $\mathbb{Z}$. Consider this pair of conic sections: \begin{eqnarray*} a^2 + 2b^2 - 3c^2 - 6d^2 &=& 1 \\ ab - 3 cd&=& 0 \end{eqnarray*} ...
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2answers
32 views

Conic sections & eccentricity: known vertical hyperbola, find radius of corresponding circle?

It is known that the eccentricity of a hyperbola is >1, and that of a circle is 0. These are both conic sections of (the lower nappe of) a (double) cone, with r the radius and h the height of the cone,...
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0answers
26 views

What are vertices and foci in hyperbolas?

Hi my class is covering conic sections and i have been introduced to hyperbolas. I couldnt get the meaning of vertices and foci. Please help explaining or clear things out sorry and thanks!!!
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1answer
24 views

Specific method for finding focus of oblique parabolas?

Its easy to find focus of standard parabolas but how to find directrix,focus,vertex etc. of oblique parabolas(Whose axis is not parallel to x or y axis).Any help:)
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0answers
21 views

Finding a mapping from a segment of the real line to an ellipse that satisfies the given property.

Here's the exact problem statement as it was given: Consider the ellipse $x^2 + a^2y^2 = 1$ Find a holomorphic isomorphism from a neighborhood of 0 onto some neighborhood of $a^{-1}i$ which transforms ...
9
votes
2answers
199 views

Is this (conjectural) geometric property intrinsically related to the distribution of Primes?

Given the series of prime numbers greater than 9, we can organize them in four rows, according to their last digit ($1,3,7$ or $9$). The column in which they are displayed is the ten to which they ...
1
vote
4answers
42 views

How do I find the equation of a tangent to a hyperbola whose centre is (h,k)?

Given that $\frac{(x-3)^2}{9} - \frac{(y-2)^2}{4} = 1$ is equation a hyperbola, I have to find its tangent at the point $\left(-2,\frac{14}{3} \right)$. I know about the equations $c^2=(am)^2-b^2$ ...
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votes
2answers
48 views

Finding the focus and directrix of the parabola $x^2=-8y$ [closed]

If the equation of a parabola is $x^2 = -8y$. Find the coordinates of the focus and the equation of the directrix. I don't understand what "coordinates of the focus" means.
1
vote
1answer
40 views

What are some great geometric properties of a rectangular hyperbola?

I have seen that ellipse and hyperbola have a lot in common. One thing that is bugging me is the fact that I know a lot of the special case of ellipse where the major and minor axes are equal (circle) ...
4
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0answers
149 views

Constructional proof of ellipse property

I came across the fact that the following function defines a family of ellipses with focal distance $f$, parameterized by the value of the function: $$\frac{x-f+\sqrt{(x-f)^2+y^2}}{x+f+\sqrt{(x+f)^2+...
3
votes
0answers
51 views

Why is euclidean geometry also called parabolic geometry?

Given that the three fundemental geometries are euclidean geometry (zero curvature), riemannian geometry (positive curvature) and lobachevskian geometry (negative curvature), I am curious as to what ...
0
votes
2answers
97 views

Minimise the area of an origin-centred ellipse with the constraint that it most enclose a particular circle [on hold]

If the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$ is to enclose the circle $x^2+y^2 = 2y$, what values of $a$ and $b$ minimise the area of the ellipse?
3
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0answers
55 views

When was discovered the nine point conic?

I wonder when was discovered the nine point conic. Wikipedia english article about it https://en.wikipedia.org/wiki/Nine-point_conic is misleading. The nine point conic wasn't discovered in 1892. In ...
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votes
2answers
38 views

Arc length of a point on ellipse from the vertex

How is the arc length of an ellipse (measured from the vertex) defined by $x = a \cos (\theta)$, $y = b \sin(\theta)$ given by $s(\psi) = a Elliptic\left(\psi,\sqrt{1-\frac{b^2}{a^2}}\right) $. Please ...
0
votes
0answers
28 views

how does the arc-length of a parabola exposed to water change when rotating the parabola with a fixed volume of water inside?

this is a generic question, but say you have a parabola with water (symbolizing a wineglass), will the arclength exposed to this fixed volume of water change as you rotate the parabola?