Questions tagged [conic-sections]
For questions about circles, ellipses, hyperbolas, and parabolas. These curves are the result of intersecting a cone with a plane.
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Find the area of the region containing the points inside the ellipse $2x^2+y^2-2xy-4x=0$ but outside the ellipse $2x^2+y^2+2xy-4x=0$.
Find the area of the region containing the points inside the ellipse
$2x^2+y^2-2xy-4x=0$
but outside the ellipse
$2x^2+y^2+2xy-4x=0$.
I was able to find the required area using integration. But it is ...
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What are these conics invariant under linear maps?
Let
$$A=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$$
be a matrix with determinant $1$. Then one can see that the conic given by the equation
$$Q(x,y)=cx^2+(d-a)xy-by^2=C,\quad C\geq 0$$
is ...
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Intersection of cones and planes
I need to calculate the volume of the region bound by :
The cone $z^2=x^2+y^2$
The plane $z=2x+2y-2$
The plane $z=4$
I have already tried setting up a triple integral but I am having some problems ...
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Can transverse axis of a hyperbola be a function of $\theta$
A hyperbola, having the transverse axis of length 2 sin $\theta $
, is confocal with the ellipse $3x^2 + y^2 =12$
then, its equation is ?
My question is about the transverse axis as a function of $\...
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4
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An unusual equation for an ellipse
Take any line in the plane expressed parametrically as
$p(t)=(x_0,y_0)+t(1,m)=(x_0+t,y_0+tm)$.
Show that the curve given by
$q(t)=\frac{2p(t)}{1+\left|p(t)\right|^2}=\frac{(2 x_0 + 2t, 2 y_0+2tm)}{1+...
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There is a compass-like tool that can draw $y=x^2$ on paper. Is there one for $y=x^3$?
Is there a tool that can draw $y=x^3$ on paper?
I'm referring to low-tech tools, e.g. not computers.
I only know of tools that can draw $y=x^2$. The YouTube video "Conic Sections Compass" ...
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Condition for angle between two lines less than 90° [closed]
I have asked to find the condition such that the lines joining focii of an ellipse don't subtend right angle at any point on ellipse.. pllz tell me the condition which i have to apply btw the two ...
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Why do the conic sections differ in these two envelopes?
I graphed a square with sides:
$y=x$
$y=-x+40$
$y=x+40$
$y=-x$
Then I divided each side to 20 equal parts and drew lines through points:
$(1,39)$ & $(19,19)$: (line 1)
$(2,38)$ & $(18,18)$: ...
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True or false: For every $n\in\mathbb{Z^+}$, there exist $a,b,c$ such that $y=(x-a)^2+b$ and $y=c$ enclose exactly $n$ lattice points.
It is easy to show that, for every $n\in\mathbb{Z^+}$, there exists a circle that encloses exactly $n$ lattice points (points with integer coordinates). Can we say the same thing about a parabola and ...
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How to draw a parabola using basic equipment?
A straight line can be drawn with a straightedge.
A circle can be drawn with a compass.
An ellipse can be drawn with string and pins.
How can we draw a parabola, using basic equipment?
Remarks
The ...
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Geometry of underdetermined regression loss function in $n$-dimensions
Let's say I have a regression loss function defined as $(AX-y)^2$, where $A \in \mathbb{R}^{m \times n}$, $X \in \mathbb{R}^{n \times 1}$ and $y \in \mathbb{R}^{m \times 1}$
If $A$ is well determined, ...
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Let $E$ be an ellipse in the plane. How can we construct an equilateral triangle whose vertices are in E? [closed]
Can an equilateral triangle be constructed from an arbitrary point $A$ on the ellipse such that that the other two vertices are on the curve?
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Geometric construction of an ellipse enscribed within a in irregular quadrilateral
I am trying to construct the faces of a cube in 3 point perspective, with ellipses enscribed in the same way as shown in this post. I can only use a straight edge and compass, but I can construct an ...
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Locus of the focus of an ellipse sliding along the coordinate axes? [duplicate]
I approached this problem by taking the centre be points p,q, I know the locus of the centre of the ellipse by the director circle property and added lengths aecos(theta) and aesin(theta) to the ...
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Three properties of a parabola with three tangents
About ten days ago I discovered three beautiful properties of a parabola with three tangents drawn using the GeoGebra program. I would like to get proof of these properties and also know if any of ...
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Normalize speed of parametric function
I have a parametric function for an ellipse:
$$f\left(t\right)=\left(a\cos\left(t\right),b\sin\left(t\right)\right)$$
As the function goes linearly through t from 0 to 2pi, the point speeds up near ...
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Scaling an Ellipse to be Tangent to Another Ellipse with a Different a/b Ratio
I need to scale an ellipse to be tangent to another ellipse without moving them from their centers $(h_1,k_1)$ $(h_2,k_2)$ or modifying their a/b ratio.
$$\frac{(x-h)^2}{(a*s)^2} + \frac{(y-k)^2}{(b*s)...
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How to represent the co-ordinates of a point on a curve without using root [duplicate]
I get these issues while dealing with curves, for example If i want to represent a point on a parabola $ax^2+by^2+2hxy+2gx+2fy+c=0$ $(h^2=ab)$in the form $(p,q)$, if I use ($p,\sqrt{q \text{ with ...
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Ellipses inscribed in parallelograms $y=\pm 1$, $y=m(x\pm1)$.
Lately I saw many questions about ellipses inscribed in quadrilaterals in Math Stack Exchange.
By substituting the equations of lines in the general equation
$$ax^2+bxy+cy^2=1$$
of an (maybe) ellipse ...
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Prove that a tangent line at a point X of a parabola bisects the angle between the point vector and axis vector
I'm trying to solve excercise $14.7.14$ in Tom Apostol's Calculus vol. $1$. I started with a parabola $y=\frac{cx^2}{2}$, where its focus is at point $(0, \frac{c}{2})$.
Then, expressed as a vector ...
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Tangent to hyperbola
Suppose we take a standard hyperbola, symmetric about the origin. Then the part of hyperbola in the 1st quadrant would be a mirror image of the part in 2nd quadrant with y axis as mirror; and the part ...
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Primes $(a,b,c)$ such that $a^2+b^2+c^2-ab-ac-bc=28$?
This problem was the first version of this one which was simplified by the O.P. to its actual version.
For what prime values of (a,b,c) the expression $f(a,b,c)=a^2+b^2+c^2-ab-ac-bc$ is equal to 28?.
...
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Parametric point on hyperbola
The parametric point on hyperbola is $(a\sec\theta, b\tan\theta)$.
Why is that when the parametric angle $\theta$ is in 1st quadrant, it represents the part of hyperbola in 1st quadrant, and similarly ...
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Arc Length of the parabola
Can You please tell me how to derive this formula?
$$L=\frac12\sqrt{b^2+16a^2} + \frac{b^2}8a \ln \left(\frac{4a+\sqrt{b^2+16a^2}}b\right)$$
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Reference for modern synthetic geometry
I have a great interest in synthetic geometry since I was in senior high school. Although I have graduated from university I still like them now. When I was in high school, I read many textbooks and ...
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Families of orthogonal curves to parabola.
In a typical ODE course, we learn that families of orthogonal curves to parabola $y=Ax^2$ are given by families of ellipses, given by $x^2/2+y^2=c^2$. However, there is this thing called parabolic ...
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Exercise 5.4.1 in Smith's Invitation to Algebraic Geometry; lines that are tangent to conic are closed subvariety of Gr$(2,3)$ in $\mathbb{P}^2$
I am trying to solve Exercise 5.4.1 in Karen E. Smith's Invatation to Algebraic Geometry:
Fix an irreducible conic $C$ in $\mathbb{P}^2$. Show that the set of lines in $\mathbb{P}^2$ that fail to ...
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Motivation for conic section ig?
Recently I've become familiar with conics as my high school academics include this topic. My textbook has a brief discussion where I learned how circle, ellipse, parabola and hyperbola are defined ...
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Find the equation of the locus of a point the difference of whose distances from two fixed points is constant given their coordinates.
So the fixed points are $$F_1=(p_1,q_1)$$$$F_2=(p_2, q_2)$$
Mid-point of foci(centre) is $$\left(\cfrac{p_1+p_2}{2},\cfrac{q_1+q_2}{2}\right)=(c_x,c_y)$$ and the the point $P=(h,k)$
The equation is ...
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Minimizing perimeter and area of the section obtained by intersection of a given line passing through a point, and the conic.
Suppose I have any second degree conic and a point $P$ lying within the conic. A family of lines pass through this point $P$. We have to find the line for which the area of the corresponding section ...
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Find all inscribed ellipses in a given triangle passing through two given internal points
Given a triangle, and two points inside it, I want to determine all the ellipses that are inscribed in the triangle and passing through both of the two given points.
My attempt: is outlined in my ...
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Find all inscribed ellipses in a given convex quadrilateral passing through a given internal point
Given a convex quadrilateral, and a point inside it, I want to find all ellipses that are inscribed in the quadrilateral and passing through the given point.
My attempt: is outlined in my solution ...
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Determine the circle that is tangent to three given ellipses
Given three ellipses in the plane specified as follows
$(r - C_1)^T Q_1 (r - C_1) = 1$,
$(r - C_2)^T Q_2 (r - C_2) = 1$
$(r - C_3)^T Q_3 (r - C_3) = 1$
I want to find the circle that is externally ...
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derivative of equation for horizontal parabolas
the equation for vertical parabolas makes sense as the tangent at the vertex of vertical parabolas has a slope of zero,
so derivative of $y=ax^2+bx+c$ which is $2ax+b$ should be 0 when we want to find ...
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Proving that the Concentric Circle Method constructs an Ellipse
I'm in my second year of an engineering drawing course, and today our professor taught us how to construct an ellipse by a method called the Concentric Circle Method (steps shown below).
My question ...
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Determining the ellipse tangent to the sides of a given convex pentagon [duplicate]
Question: Given the five vertices of a convex pentagon, determine the ellipse that is tangent to the sides of this pentagon.
For example, if the five vertices are: $P_1 (11,0) \ , P_2(15,0) \ , P_3 (...
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Why is there no $\pi$ in an ellipse?
Allow me to clarify ...
With a circle of circumference $c$ and diameter $d$, $\pi$ makes an appearance as $\frac{c}{d}$ even though the equation of a circle, $(x - h)^2 + (y - k)^2 = r^2$, doesn't ...
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Why does the intercept form of the straight line equation $x/a+y/b=1$ look similiar to the standard equation of an ellipse $x^2/a^2 +y^2/b^2 =1$ [closed]
Today we started 2-D geometry in school covering straight lines and conic sections. And I just had this shower thought, why do these two standard equations look similiar. Is there any implication of ...
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Show that $y=\frac{1}{x}$ is a hyperbola
I am trying to write $y=\frac{1}{x}$ in the standard form for a hyperbola, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ ($a>0$, $b>0$) but I can't figure out the last step. My book says to
First, rotate ...
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Tilted Parametric Ellipse to Lissajous
Given that I know the values of the parametric equation of a tiltet ellipse:
$$x(α)=R_x\cos(α)\cos(θ)−R_y\sin(α)\sin(θ)$$
$$y(α)=R_x\cos(α)\sin(θ)+R_y\sin(α)\cos(θ)$$
with $R_x,R_y$ as the mayor/minor ...
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A circle is described on AA',major axis of ellipse(diameter).For point P on circle:AP,A'P joined cutting ellipse at Q,Q'. (AP/AQ)+(A'P/A'Q')=3. Find e
A circle is described upon $AA'$, the major axis of an ellipse as diameter.
P is a point lying on the circle.
Let $AP,A'P$ be joined cutting ellipse in $Q,Q'$
It is given that $\dfrac{AP}{AQ}+\dfrac{A'...
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votes
1
answer
55
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$2$ conics intersect at $2$ real points, geometric construction on $\Bbb R^2$ of the line through 2 non-real points?
On $\Bbb R^2$ suppose two conics intersect at two points, then they intersect at two other non-real points on $\Bbb C^2$, it is proved in the comment that the line through the two non-real points is ...
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Help resolving an apparent contradiction with triangle internal angles
I've been scratching my head over this for the last few days...
I've been doing some maths relating to orbital mechanics and elliptical orbits, and there's one simple thing that I can't get right. I ...
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Maximum of a constant for which inequality doesn't have solutions
I have a problem with a system of non-linear inequalities:
\begin{cases}
(x-\frac{y}{B})(y-x-1) \ge \frac{(y-1)x}{2} \\
y \ge 2
\end{cases}
and I would like to find the exact value of the maximum of ...
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Equation of parabola using axis and tangent at vertex
Let $L=ax+by+c=0$ be the axis of a parabola, and the tangent at vertex be $M=bx-ay+d=0$
If $L^2=4pM$ is the equation of that parabola whose length of latus rectum is $4p$, then $a^2+b^2$ must be equal ...
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2
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Locus of midpoint of chord of an ellipse whose length is constant
Consider an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$
We want to find the locus of the midpoints of all those chords whose length is constant $(=2c)$
Here's my approach:
Let the midpoint of the ...
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1
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Angles subtended by tangents from a point on the focus of a conic section
I know that the tangents from a point to a conic section subtend equal angles on the focus.
However, I have mostly studied conic sections from the perspective of coordinate geometry, so even when ...
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Can an ellipse have integer values for perimeter and area?
Is it possible to find an ellipse whose perimeter and area are integer values? I guess that such ellipse doesn't exist, but I haven't fully proved that.
Obviously if both the semiaxes are algebraic, ...
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Relationship between conic section and angle of incidence
I am attempting to derive an equation that relates the area of an ellipse to its oblique cone angle alpha.
My knowns are the height and semi-major/semi-minor axis. My unknown is the angle alpha. ...
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Determining the minor radius of an ellipse from its major axis and a tangent line
I'm playing around with modeling an archway of a building using a solid modeler, and have come to a problem that exceeds my mathematical ability.
Given an ellipse that has a known major radius, and is ...
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