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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Let P1 = (x1, y1). Describe the set of all points P = (x,y) in R2 such that ||P-P1|| = 9 by identifying the type of conic and finding its equation.

Let P1 = (x1, y1). Describe the set of all points P = (x,y) in R2 such that ||P-P1|| = 9 by identifying the type of conic and finding its equation. I'm sorry, but this question throws me off in many ...
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2answers
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Find the equation of the locus of the mid-point between an elliptical point and its directrix

I'm struggling with this question: The point $P$ lies on the ellipse $x^2+4y^2=1$ and $N$ is the foot of the perpendicular from $P$ to the line $x=2$. Find the equation of the locus of the ...
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25 views

Perimeter of an ellipse intuition help

I am aware that you can take the circumference of an ellipse using an elliptic integral and haven't looked much into it, but that seems to be a bit extreme and i was taking a personal look at conic ...
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Hyperbola problem : From a point P(1,2) pair of tangents are drawn to a hyperbola H in which one tangent to each arm of hyperbola…

Problem : From a point P(1,2) pair of tangents are drawn to a hyperbola H in which one tangent to each arm of hyperbola. Equation of asymptotes of hyperbola H are $\sqrt{3}x -y+5=0$ and ...
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Trying to show that conic section $ax^2 + 2bxy + cy^2 = d$ takes on different shapes.

Note: This is a homework problem. I'm trying to show that conic section $ax^2 + 2bxy + cy^2 = d$ is an ellipse or the empty set if $ac-b^2\gt 0$. There are others to show but if I can understand this ...
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Find the minimum radius of the circle which is orthogonal to two given circles

Problem : Find the minimum radius of the circle which is orthogonal to both the circles $x^2+y^2-12x+35=0$ and $x^2+y^2+4x+3=0$ . Solution : Let the equations : $x^2+y^2-12x+35=0.....(i)$ and ...
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Find the equation of the parabola with its vertex on the line $2y-3x=0$?

Its axis of symmetry is parallel to the x-axis, and it passes through the two points $(3,5)$ and $(6,-1)$
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If the length of tangent drawn from an external point P to the circle of radius $r$ is $l$ , then prove that area…

Problem : If the length of tangent drawn from an external point P to the circle of radius $r$ is $l$ , then prove that area of triangle form by pair of tangent and its chord of contact is ...
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1answer
48 views
+150

Defining ellipse using points and normal vectors from them

There is an article on how to detect circles in images using pairs of gradient vectors (assuming the circle is dark and background is bright). The thing is that gradient of image intensity at each ...
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42 views

Asymmetric hyperbola-type curve? (for fitting to data)

I have this question: what would be the name and equation of a curve which resembles a parabola but has not the requirement of symmetry? So the general parabola equation is: $y=ax^2+bx+c$ I must ...
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1answer
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Finding angle of a spigot that produces a parabolic fountain of water

I am currently doing a math exploration and I need help in determining how to find the angle of a spigot that would maximize the area under a parabolic fountain. I thought of this topic to ...
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2answers
36 views

Volume, Lateral Area, and Surface Area of an Elliptic Conical Frustum

What are the formulae for the volume, surface area, and lateral area (i.e. the surface area without the bases) for the above illustrated elliptic conical frustum? I think I've got the volume figured ...
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1answer
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Conic sections in standard form

I'm trying to convert the equation $$x^2 +2y^2 +4x-4y+4=0$$ into its standard form by choosing a new set of axes. Yet, when I go down the conventional route, there is no xy term so ...
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1answer
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a problem with Stokes' theorem(curl)

If L is the circle which you get from the intersection between the sphere $$ x^2+y^2+z^2=1, y=x\sqrt(3) $$ and $$ I= \int_L (y-z)dx+(z-x)dy+(x-y)dz $$ so |I| equals to? but i dont understand how the ...
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1answer
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Parabola max. number

If the directrix and the tangent at vertex of a parabola are given then what is the maximum number of parabolas that can be drawn? Well according to me the answer should be 1 because the distance ...
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1answer
78 views
+500

Prerequisites for Appolonius Conics?

I want to get Thomas Heath's version of Apollonius's Conic Sections. Does anyone know the prerequisites to understand everything in this book? I heard I would need the Euclid's Elements book on Solid ...
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Roulette of a parabola - Delaunay-Surface

I've problems to understand an equation I've found in various books and papers. Maybe someone could help me and explain it a little bit more precisely. I colored the equation in yellow (the picture ...
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Find the number of points on ellipse $\frac{x^2}{50}+\frac{y^2}{20}=1$ from which pair of perpendicular…

Find the number of points on ellipse $\frac{x^2}{50}+\frac{y^2}{20}=1$ from which pair of perpendicular tangents are drawn to ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ Normal from a point ...
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3answers
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If A represents area of ellipse $3x^2+4xy +3y^2=1$ then the value of $\frac{3\sqrt{3}}{\pi}A = ?$

If A represents area of ellipse $3x^2+4xy +3y^2=1$ then the value of $\frac{3\sqrt{3}}{\pi}A = ?$ My approach : Since area of ellipse is $\pi ab$ where a is semi major axis and b is semi minor ...
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Ellipse 1 : E$_1 = \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 ( a > b)$ another ellipse 2 : $E_2$ which passes thru…

Problem : Ellipse 1 : E$_1 = \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 ( a > b)$ another ellipse 2 : $E_2$ which passes thru extremities of major axis of $E_1$ and has its foci at ends of its minor axis. ...
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In an equation of a Hyperbola, what is the relation of the a and b terms?

I am studying hyperbolic equations (with conic sections) in pre-calculus algebra. I am a bit confused about the order of the a and b terms in the equation for a hyperbola with center at (h, k): ...
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1answer
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Equation of a parabola with vertex $V$ and point $P$

Find the equation of the parabola which has the given vertex $V$, which passes through the given point $P$, and which has the specified axis of symmetry. $V(4,-2), P(2,14)$, vertical axis of ...
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3answers
26 views

How to determine family of circles passing through two given points?

The question asks to show that the equation of any circle passing through two given points takes a certain form. I have obtained the points as being $(2,1)$ and $(2,-1)$ but I'm not sure as to how to ...
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2answers
81 views

Find all natural number solutions to: $20x^2 + 11y^2 = 2011$

I believe that the equation $20x^2 + 11y^2 = 2011$ describes an ellipse. I don't know how to solve for the $x,y \in \mathbb{N}$ that satisfy this equation.
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1answer
61 views

shape created by parabola

What would be the name of the shape that is the set of all points such that they are equidistant from the point $(0,1)$ and to the parabola $y=x^2$. Here is a desmos graph that generates the ...
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1answer
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Find the number of common tangents to $y^2=2012x$…

Problem : Find the number of common tangents to $y^2=2012x$ and $xy =(2013)^2$ Solution : Common tangent will have slope equal to both curves. therefore, differentiation both the curves we get ...
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0answers
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P is a point t on the parabola $y^2=4ax$ and PQ is a focal chord. PT is a tangent at P and QN is a

Question : P is a point t on the parabola $y^2=4ax$ and PQ is a focal chord. PT is a tangent at P and QN is a normal at Q. Find the minimum distance between PT and QN. Solution : Since the ...
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An equation of a tangent to the parabola $y^2=8x$ is y=x+2. the point on this line from which the other tangent

Problem : An equation of a tangent to the parabola $y^2=8x$ is y=x+2. the point on this line from which the other tangent to the parabola is perpendicular to the given tangent is given by ... ...
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The point of intersection of the tangents to the parabola $y^2=4x$ at the points where the circle $(x-3)^2+y^2=9$

Problem : The point of intersection of the tangents to the parabola $y^2=4x$ at the points where the circle $(x-3)^2+y^2=9$ meets the parabola, other than the origin, is .. Solution : Point of ...
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1answer
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The tangent at the point $P(x_1,y_1)$ to the parabola $y^2=4ax$ meets the parabola $y^2=4a(x+b)$ at Q and R,

Problem : The tangent at the point $P(x_1,y_1)$ to the parabola $y^2=4ax$ meets the parabola $y^2=4a(x+b)$ at Q and R, the coordinates the mid point of QR are ? Solution : Tangent from a point ...
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1answer
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A circle touches the parabola $y^2=4ax$ at P. It also passes through the focus S of the parabola and int…

Problem : A circle touches the parabola $y^2=4ax$ at P. It also passes through the focus S of the parabola and intersects its axis at Q. If angle SPQ is $\frac{\pi}{2}$ find the equation of the ...
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2answers
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Trains describing a parabola

From the train station – the point S – originante two tracks, i.e. rays, which do not lie on a common straight line. Along these move two trains, which are line segments. On the first track a train is ...
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1answer
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Theory of tangents and normals of an ellipse

What are the number of distinct normals that can be drawn to an ellipse from a point inside ,on and outside an ellipse?
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1answer
32 views

Geometric or analytic proof that in hyperbola, $c^2=a^2+b^2$

How to prove (geometric or analytic) that in hyperbola $c^2=a^2+b^2$? Given that $a$ is the undirected distance of the center to one of the vertices, $b$ is the undirected distance of one of the ...
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What is the best method to find common tangents to two second degree curves?

Often I found myself making my life hard finding common tangent to second degree curves, for example what I would do in case of the following example: Find common tangents to ...
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Find coordinates of vertex and equation of parabola [duplicate]

A parabola has its focus at $(0,4)$. Its directrix is $y = -4$. Q) Find the coordinates of the vertex and write the equation of the parabola. Thank you!
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How to draw an ellipse, given its center, its major radius and two arbitrary points from its perimeter?

I'm trying to write as practice a program where I could visualize a circle being freely moved around in space, and long story short is that I ended up with this problem that I could not solve Besides ...
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1answer
28 views

Use the pseudoinverse to find the conic section of best fit to the data

I am working on a group project and none of us can figure out how to find the answer. Our professor insists that all of our work be done in maple. The problem is: Use the pseudoinverse to find the ...
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1answer
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How many non-collinear points determine an $n$-ellipse?

$1$-ellipse is the circle with $1$ focus point, $2$-ellipse is an ellipse with $2$ foci. $n$-ellipse is the locus of all points of the plane whose sum of distances to the $n$ foci is a constant. I ...
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Parabola - How far from the thrower does the ball strike the ground?

The height of a ball thrown in the air is given by $h(x) = \frac {–1}{12} x^2 + 6x+ 3$, where x is the horizontal distance in feet from the point at which the ball is thrown. c. How far from the ...
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Find the locus of points whose distances from the line $y=\sqrt3x$ and x-axis are equal.

Find the locus of points whose distances from the line$\hspace{0.2cm}$ $y=\sqrt3x$$\hspace{0.2cm}$ and x-axis are equal. My solution:I start with the following ...
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1answer
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Parabola investigation

Edit 4: I added the below picture for clarity I'm trying to figure out how to find the angle between the red line and the blue line, but I have no idea how to start. (I have a feeling that this ...
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3answers
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Using trig identities to change from parametric to Cartesian equation

$$x=\sin t\\ y= 3\cos (3t)$$ Find $y$ in terms of $x$. I have graphed the function and it appears to follow $y(x)=-4x^2 +2$ from $-1\le x\le 1$ and $-2\le y\le 2$ . Thanks
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What's the parametric equation for the general form of an ellipse rotated by any amount? Thanks

What's the parametric equation for the general form of an ellipse rotated by any amount? Preferably, as a computer scientist, how can this equation be derived from the three variables: coordinate of ...
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Conic sections and common functions

Is there a intuitive proof/reason of why plots of some common functions like y=x^2 are shaped like cross sections of a seemingly unrelated 3D object like a cone? ...
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I need some ideas regarding working model in mathematics. The topics are conics and vectors.

Hi can someone suggest me some working mathematical models under the topics conics and vectors. It should be 12th standard level. (Higher Secondary). I am trying to help my juniors to do this project. ...
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1answer
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Parabola equation expressed after x

Sorry for the bad title, as English is not my main language. Let me explain better what I mean. I have this equation of parabola: $y = x^2 + 4x $ What I want to do is get the $x$ in one side and ...
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1answer
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Polar correlation and conics in $\Bbb RP^2$

I'm stuck on a small detail in Proposition 1.2.8 in Geiges' Introduction to Contact Topology. Let $C$ be a conic in $\mathbb{R}P^2$ given by $q^tAq=0$, where $A$ is a nonsingular, symmetric 3x3 ...
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1answer
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Find the parametric equation of the following parabola?

It doesn't give me $2$ equations this time just $1$ and I have no clue what to do; $y^2 = 4x$ ANSWER IN BOOK: $x = t^2, y = 2t$
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Ellipsoidal Decomposition: Finding ellipsoids whose sum contains a given ellipsoid

We have a known ellipsoid $E\left(q,Q\right)$ in a 2D space. $q$ represents the center of the ellipsoid and $Q^{-1}$ is the weight matrix. The general equation of the ellipsoid is given as: ...