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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.

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The normal at $T(at^2,2at)$ of parabola $y^2=4ax$ meets the parabola again at $S(as^2,2as)$. Show that $t^2+st+2=0$.

The normal at the point $T(at^2,2at), t\not = 0$, on the parabola $y^2=4ax$ meets the parabola again at the point $S(as^2,2as)$. Show that $t^2+st+2=0$. I am completely lost. I tried using implicit ...
1
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6answers
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Find an equation of a tangent at $C(3,1)$ on $x^2-y^2 = 8$ with an elementary method of analytical geometry.

Find an equation of a tangent at $C(3,1)$ on $x^2-y^2 = 8$ with an elementary methods of analytical geometry. So with non calculus method! The focuses are at $A(4,0)$ and $B(-4,0)$. It is well known ...
2
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1answer
41 views

Parabola - Definition as a locus of points

On Wikipedia, a parabola is defined as follows: A parabola is a set of points such that, for any point $P$ of the parabola, the distance $|\overline{PF}|$ to a fixed point $F$, the focus, is equal ...
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0answers
49 views

Let $E$ is ellipsoid in $\mathbb{R}^n$.

Let $E'$ is ellipsoid of dimension $n-1$ that gain as intersection of $E$ and some hyperplane. Let $a_1\leq\cdots\leq a_n$ are halfaxis of $E$ and $b_1\leq\cdots\leq b_{n-1}$ are halfaxis of $E'$. ...
3
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2answers
34 views

I'm stuck on coordinate graph involving tangents of parabolas…

Consider the function $f(x) = \max \{-11x - 37, x - 1, 9x + 3\}$ defined for all real $x.$ Let $p(x)$ be a quadratic polynomial tangent to the graph of $f$ at three distinct points with $x$-...
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0answers
22 views

Conic in Trilinear Coordinates

I have the following equation of a conic in trilinear coordinates: $$x^2+y^2+z^2-\frac{\alpha^2+\beta^2}{\alpha\beta}xy-\frac{\beta^2+\gamma^2}{\beta\gamma}yz-\frac{\gamma^2+\alpha^2}{\gamma\alpha}zx=...
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4answers
25 views

Quadratic equation and two points.

I need to solve a quadratic equation (actually I need to explain it to my kid), but I get stuck in the middle and would be grateful, for any pointers into the right direction. $y=ax^2+bx-1$ with two ...
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0answers
11 views

How to find the distances between P0 to P1 in this 3 dimensional ellipsoid? How to find distance b/w two foci? [on hold]

(Kindly think your idea before looking at details file ) 1. How to find the distances between P0 to P1 in this 3 dimensional ellipsoid? Can give example please ? 2. How to find the distance D between ...
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1answer
49 views

Limits for integral over ellipse

How do I find the limits when trying to integrate over an ellipse? (1) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ Edit: I'm trying to find the area of the part of the plane $Ax + By +Cz = D$ lying ...
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1answer
22 views

How to make reparameterisation of ellipsis $c(t)=(2\sin t,\cos t)$? [on hold]

How to make reparameterisation of ellipse $$c(t)=(2\sin t, \cos t)?$$
1
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1answer
42 views

a circle and a parabola have 3 intersection points

Is it possible that a circle and a parabola on a euclidean plane have 3 intersection points and the center of the circle does not lie on the axis of parabola?
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0answers
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Approximate Tangent Line of Hyperbola

Let us say that the hyperbola $$ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$ can be written as $$ y^2=\frac{b^2}{a^2}(x^2-a^2) $$ can I approximate the tangent line at a point $(x_1,y_1)$ as $$ y = \frac{b}{...
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0answers
25 views

P is any point on the ellipse whose focus are S, S' then wrt triangle SPS' [closed]

P is any point on the ellipse whose focus are S, S' then wrt triangle SPS' A) the ex centre opposite to side SS' lies on tangent at P B) the ex centre opposite to side SP lies on normal at P C) the ...
3
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0answers
69 views

Finding center and rotation angle of ellipse that contains three points

Given three points $p_1, p_2, p_3 \in \mathbb{R}^2$, and an ellipse with shape parameters $(a,b)$ (the semi-major and semi-minor), is it possible to determine, if they exist, a center $c \in \mathbb{R}...
2
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1answer
34 views

Proving Frégier's Theorem for conics

I am looking for a proof of Frégier's theorem for conics. Pick any point $P$ on a conic section, and draw a series of right angles having this point as their vertices. Then the line segments ...
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2answers
32 views

Find all integer solutions to hyperbola.

The question is find all integer solutions to $6xy+4x-9y-7=0$ Now I did trial and error and found $(1,-1)$ to be an integer solution. This question is in my summer math homework. We have not been ...
3
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3answers
58 views

Polar form of a conic section

Could someone show me how to find a polar form of this general equation of a conic section? $x^2+y^2-xy+x=4$ I have managed to determine this is an ellipse and write it in a canonical form with ...
3
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3answers
51 views

Find the equation and height of an elliptical whispering room

The room is 150 feet long and the distance from the center of the room to the foci is 60 feet. Finding $a^2$ is easy its $$2a=150$$ $$a=75$$ $$a^2=5625$$ but where I get lost is finding $b^2$, I ...
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2answers
43 views

Using any parabola, find $𝑐$ such that it is tangent to the line $𝑦=𝑥$.

Using any parabola, find $c$ such that it is tangent to the line $y=x$. I am not sure what this question means. I thought tangent lines were straight? How can a parabola like $x^2 + x + 1$ be "...
7
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2answers
282 views

2 parabolas through 4 points

I'm looking for a nice projective solution for a problem: "why can we draw exactly 2 parabolas through 4 points (which are located in vertices of a convex quadrilateral) on the real affine plane?" ...
1
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3answers
22 views

Find a rational function $f(x)$ with H. asymptote of y=2, V. asymptotes at x=-3, x=3 and a y-intercept at $\frac{-2}{3}$

So first I multiplied the V. Asymptotes like so, $$(x+3)(x-3)$$ to get $$(x^2-9)$$ And knowing that because the horizontal Asymptote is a non-negative number, that the leading coefficient in the ...
6
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2answers
197 views

Given four points, determine a condition on a fifth point such that the conic containing all of them is an ellipse

The image of the question if you don't see all the symbols The given points $p_1,p_2,p_3,p_4$ are located at the vertices of a convex quadrilateral on the real affine plane. I am looking for an ...
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0answers
29 views

Calculating the asymptotes of an hyperbola from the general equation [duplicate]

Suppose that the conditions which ensure that the following equation is an hyperbola are verified: $$ax^2+2bxy+cy^2+2dx+2ey+f=0$$ What are the general expressions of the two asymptotes of the ...
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1answer
60 views

Why are these angles equal?

-see picture above- (from Billiards and Geometry by Serge Tabachnikov) I don't understand why the angles $F_2BA_1$ and $F_1BA_0$ are equal (I do understand the conclusion, that follows from the ...
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0answers
21 views

Find eccentricity and length of semi transverse axis of given hyperbola

Find eccentricity and length of semi transverse axis of given hyperbola? y=x-1/x or we can write x^2 - xy=1
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0answers
29 views

How to proceed in this question? I am not able to understand how $u^2$ can be related to the area?

Determine the equation of the ellipse centered at $(0,0)$ whose focal length is $8\sqrt(6)$ and the area of a rectangle in which the ellipse is inscribed within is $80u^2$.
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1answer
24 views

Rotation of a non rectangular hyperbola: equation of hyperbola referred to its asymptotes

I'm looking for a way in which I can rotate a non rectangular hyperbola; in particular I'd like to get the equation of a non rectangular hyperbola referred to its asymptotes. To do it I need to ...
0
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1answer
40 views

Ellipse: Known Distance from Focus to Far Side $(A+C)$ and $B$

I have a problem where I know the distance from one of the foci to the far side of the ellipse $(A+C)$ and I know $B$. How would I find out what $A$ and $C$ are separately? EDIT: Sorry for the ...
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2answers
28 views

Finding the inverse function of a quadratic function [closed]

let $f:[-4,∞) \rightarrow \mathbb{R} , f(x)=-(x+4)^2 +3$. show that $f^{-1}:(-∞,3] \rightarrow \mathbb{R}, f^{-1}(x)=\sqrt{3-x}-4.$ a question from my 11th-grade maths assignment. I don't even know ...
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1answer
53 views

Finding the center, radius of a circle when there is a constant in front of the variables [closed]

$$4x^2+(y-2)^2 = 4$$ How to find the center and radius of a circle when there is a constant in front of the variables?
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1answer
38 views

Converting $\frac{x^2}{4}+y^2\leq1$ to parametric form

I have $\dfrac{x^2}{4}+y^2\leq1$ to be converted to parametric equation. I have tried, $x^2+4y^2\leq4$ $x^2\leq4(1-y^2)$ $x^2\leq4(1-y)(1+y)$ I am doubting for my next step since this is an ...
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1answer
37 views

Getting the minimum radius of curvature of a conic section

I am fairly new to this forum and since I am not directly from a mathmetics background I recently ran into a problem I cannot solve. What I am trying to do is to intersect a cone at a specific angle ...
0
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1answer
47 views

If a point lies on a conic section, its polar is the tangent through this point to the conic section

I found this fact: "If a point lies on a conic section, its polar is the tangent through this point to the conic section" here: https://en.wikipedia.org/wiki/Pole_and_polar Unfortunately I couldn'...
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0answers
25 views

Area between a parabola and a line

Hello, can someone explain me, or at least give me some instructions, in order to understand why is the surface of the arc given by that integral? I've done some research and nothing that i do is ...
1
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1answer
94 views

Find the angle of rotation and minor axis length of ellipse from major axis length, center, and two points?

I'd like to describe the ellipse centered at the origin with a fixed major axis length of $2a$ that passes through two points $(u_x, u_y)$ and $(v_x, v_y)$ (both within $a$ of the origin). In ...
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0answers
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Question on proof of 'A quadratic cone is degenerate iff it consists of all vectors, or two vector planes, or one vector plane, or one vector line'.

I'm not going to write down the whole proof. In the proof we've already showed that the quadratic cone $K$ is the union of vector planes through $\langle v \rangle$ (= vector line consisting of ...
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1answer
53 views

co-ordinates for centre of known ellipse tangent to known circle

Does any body have the equation for calculating the co-ordinates for the centre of a known ellipse tangent to a known circle 1 Sketch of ellipse and circle attached the 20 degree dimension will be ...
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2answers
26 views

Closest Point on a Parabola to a Point Formula for X-Coordinate

I'm trying to find a formula for the x-coordinate of a point on the parabola $y = -x^2-1$ that is closest to a point (x,y). Of course I can find the y-coordinate, so that's why I'm only worrying about ...
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2answers
53 views

Maximum area of triangle inscribed in an ellipse

If a triangle is inscribed in an ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, Find the Maximum area of Triangle My try: Let $A(5\cos p, 4\sin p)$, $B(5\cos q, 4\sin q)$ and $C(5\cos r, 4\sin r)$ be ...
2
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1answer
56 views

Finding equation of the parabola from tangent information

Two lines drawn through $T$ at $(-1,-2)$ are tangent to a parabola at $P$ $(2,3)$ and $Q$ $(3,-1)$, respectively. Find the equation of the parabola. I tried using similarity of triangles to find the ...
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1answer
26 views

Prove Smaller Distance from Hyperbola to Asymptote

There is a canonical hyperbola $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 $$ and the asymptote $$ y= \pm \frac{b}{a}x $$ Let us say that the value of hyperbola at $x$ is given as $$y=\frac{b}{a}\sqrt{x^2 - ...
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0answers
23 views

Maximum number of common chords that are existent between two Conics

The maximum no. Of common chords between a circle and a parabola is 6. this is because they can have at most Four Points of intersection. However I have doubt regarding other combination of conics. ...
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1answer
28 views

Extending a circle to form an ellipse by modifying the lengths parallel to axes.

Suppose that a student only has the theoretical knowledge on circles yet he's attempting a problem to include an ellipse. Is it possible to alter the coordinate axes i.e. $x$ as $x+a$ and if so are ...
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1answer
30 views

Rotate a Point on an ellipse by an angle and calculate the distance between them

I am given a starting point $S=(s_1,s_2)^T$, the Center Point $C=(c_1,c_2)^T$ of the ellipse, with major radius $a$ and minor radius $b$, also the major axis is rotated w.r.t. the X axis by an angle ...
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1answer
27 views

Distance of centre of an ellipse touching both the positive X and Y axes from origin when the ellipse is being rotated

If a horizontal ellipse touches the X-axis and Y-axis in the first quadrant, and this ellipse is rotated in anti-clockwise sense always touching the X-axis and Y-axis, till the ellipse becomes ...
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1answer
30 views

How to restrict only top range of ellipse function, and what is its domain?

I am trying to graph the function of an ellipse that is: $$1=\frac{x^2}{49}+\frac{(y+1)^2}{9}$$. I want to make the horizontal ellipse's range $y \leq 0.838$. So, when I also have to write the domain ...
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votes
3answers
60 views

derivation of ellipse parameters

At various places on the Web (including Mathematics StackExchange) are various methods of calculating the semimajor and semiminor parameters of a ellipse $(a, b)$ from the location of a focus on the $...
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1answer
25 views

How to find locus for given condition?

I've been through all the thinking but i could not get to the answer. Please help me in this. See images for details
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0answers
39 views

Area of Intersection of Two Ellipses

Given two ellipses in space where: a & c are major diameters b & d are minor diameters h & k are x-axis centers j & i are y-axis centers r & s are the rotation of each ellipse ...
23
votes
6answers
3k views

Why do early math courses focus on the cross sections of a cone and not on other 3D objects?

Conic sections seem to get special attention in early math classes. My question is why do these cross sections of cones deserve more attention than those of, say, a rectangular prism, a cube, or some ...