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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4
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3answers
209 views

Ellipse $3x^2-x+6xy-3y+5y^2=0$: what are the semi-major and semi-minor axes, displacement of centre, and angle of incline?

Given the ellipse $$3x^2-x+6xy-3y+5y^2=0$$ find the following: semi-major axis, $a$ semi-minor axis, $b$ displacement of centre from origin (or coordinates of centre of ellipse $(h,k)$) angle of ...
1
vote
3answers
128 views

Find equation of tangents to hyperbola

$$\frac{x^2}{4} - \frac {y^2}{16} = 1$$ There is a point $(1,2)$ where $2$ lines pass through and are a tangent to both curves. How do I find the equation of both lines?
1
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2answers
487 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 ...
1
vote
1answer
59 views

Find the area of triangle APB, where P is a point $(a\cos\theta, b\sin\theta)$ on an ellipse and $A, B$ are its radii points $(a,0) (0,b)$

A point $P(a\cos\theta, b\sin\theta)$ sits on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. The points $A$ and $B$ have coordinates $(a,0)$ and $(0,b)$ respectively. Show that the area of ...
0
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1answer
35 views

Let $P_1 = (x_1, y_1)$. Describe $P = (x,y)$ in $\mathbb{R}^2$ s.t. $||P-P_1|| = 9$ by identifying conic and finding its equation

Let $P_1 = (x_1, y_1)$. Describe the set of all points $P = (x,y) \in \mathbb{R}^2$ such that $||P-P_1|| = 9$ by identifying the type of conic and finding its equation. I'm sorry, but this question ...
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2answers
193 views

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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0answers
53 views

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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0answers
25 views

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 ...
3
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0answers
119 views

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 ...
0
votes
1answer
190 views

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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2answers
422 views

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
159 views

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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0answers
550 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 ...
0
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1answer
176 views

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
1k 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 ...
0
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1answer
119 views

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 ...
1
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1answer
51 views

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 ...
0
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1answer
83 views

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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2answers
172 views

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 ...
2
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0answers
78 views

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 ...
4
votes
3answers
76 views

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 ...
2
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2answers
43 views

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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1answer
24 views

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 ...
0
votes
3answers
863 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 ...
0
votes
2answers
107 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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votes
2answers
108 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
184 views

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
71 views

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 ...
0
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3answers
407 views

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 ... ...
0
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2answers
237 views

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 ...
0
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1answer
42 views

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 ...
0
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2answers
573 views

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 ...
2
votes
2answers
147 views

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
228 views

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
142 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 ...
0
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0answers
207 views

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 ...
0
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1answer
130 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
215 views

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 ...
0
votes
1answer
180 views

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 ...
1
vote
2answers
202 views

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 ...
0
votes
3answers
273 views

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
2
votes
2answers
311 views

What's the parametric equation for the general form of an ellipse rotated by any amount?

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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votes
1answer
20 views

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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vote
1answer
37 views

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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votes
1answer
39 views

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: $x = t^2, y = 2t$
3
votes
1answer
441 views

If $x^2 + y^2 + Ax + By + C = 0 $. Find the condition on $A, B$ and $C$ such that this represents the equation of a circle.

If $x^2 + y^2 + Ax + By + C = 0 $. Find the condition on $A, B$ and $C$ such that this represents the equation of a circle. Also find the center and radius of the circle. Here's my solution, ...
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0answers
132 views

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: ...
1
vote
1answer
67 views

Parabola - equation from three points

Question: Find the equation of the parabola whose axis is parallel to the y-axis and which passes through the points (0,4) (1,9) and (-2,6) Well as the parabola has its axis parallel to the y-axis ...
0
votes
2answers
263 views

Relation of ellipse semi-axes with rotation angle and projection length

In the following setup, assume $w$ (length of the projection of the ellipse) and $\theta$ (the rotation angle) are known. I want to know what equation(s) do I have here that helps me to derive the ...
2
votes
1answer
87 views

finding out wheter point is inside ellipse

I'm working on a way to determine if given point is "inside" given ellipse, the problem is I've already forgotten all the related mathematics and don't have time to relearn it and find a way to do it. ...