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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Rotated Ellipse

It is well known that the equation $$\frac {(x\cos\alpha+y\sin\alpha)^2}{a^2}+\frac {(x\sin\alpha-y\cos\alpha)^2}{b^2}=1\tag{1}$$ (where $\beta\neq\alpha$) represents an ellipse centred at the origin ...
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Finding the image in an elliptic mirror

I'm trying to find the location of the image of a point being reflected by a mirror shaped like (half of an) ellipse. The goal is to find a transformation $\mathbb{R}^2\rightarrow \mathbb{R}^2$ which ...
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How to find y of vertex of Parabola with y-intercept and x-intercepts

I already know how to find the x of the vertex with this information but I do not know how to find the y of the vertex. How can I find the y? The x intercepts are (-1,0) (5,0) and the y intercept is (...
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How is land area calculated when the ellipsoidal shape of the Earth cannot be neglected?

I was curious as to how the land area of a state such as Colorado could be calculated. I understand the area of a 2D rectangle can be calculated using the formula width times length. However, I was ...
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What are the effects when changing the values of $a$ and $b$?

The question given is: The general equations of three of the conic sections with their centres at the origin are given. Explore the effect of changing the values of $a$ and $b$. I have been able to ...
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What is the equation of the hyperbolic path?

I'm struggling with this question and any help would be greatly appreciated. Alpha particles are deflected along hyperbolic paths when they are directed towards the nuclei of gold atoms. If an ...
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1answer
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Real life applications of a circle? (Conics)

for my Math 2U assignment, we have to discuss real life applications of different conic sections. However, apart from the wheel, I cannot find or think of any other real life applications of the ...
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Find area of cross section of cylinder by the plane $x$

I am working on my scholarship exam practice (assume high school/pre-university math background) and I think I got half way through but I am not sure how I could continue. Let $r$ be a positive ...
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If $x$ and $y$ are integers such that $5 \mid x^2 - 2xy - y$ and $5 \mid xy - 2y^2 - x$, prove that $5 \mid 2x^2 + y^2 + 2x + y$.

Given that $x$ and $y$ are integers satisfying $5 \mid x^2 - 2xy - y$ and $5 \mid xy - 2y^2 - x$, prove that $5 \mid 2x^2 + y^2 + 2x + y$. I have provided a (dumbfounding) solution down below if you ...
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simple complex number proofs

Here is one question on my text book: $P$ is a point on an argand diagram corresponding to a complex number $z$ which satisfies equation $|4+z |-|4-z |=6$, prove that$$| 4+z |^2-| 4-z |^2 \ge 48$$ ...
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Determine conic given two points on the conic and equation of major and minor axis.

Is it possible to determine a Conic given two points on the conic and equation of major and minor axis? I choose $5$ random points on $\mathbb R^2$ independently. Since 5 points determine a conic, I ...
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convert elllipse conic representation to parametric representation

I have come across a pdf from cornell about ellipse fitting and in there it listed information on how to convert ellipse from conic representation to parametric representation. source:http://www.cs....
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Mean distance from the focus of an ellipse in polar coordinates

My question is about the average distance from the focus of an ellipse. If we let the equation of an ellipse in polar coordinates (centred at the focus) be $$r = \frac{\ell}{1+\varepsilon\cos{\theta}}...
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3answers
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Find projectile speed given maximum height and range.

I want to simulate a catapult throwing a rock in my computer game, but by design, I want all my units to shoot from a certain height, reach a maximum height and also hit a target that can be meters or ...
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1answer
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Why is the axis of a parabola parallel to the eigenvector of 0?

Let $\gamma$ be a generic conic in E2. \begin{equation} \begin{pmatrix} 1 & x & y \end{pmatrix} \begin{pmatrix} a_{00} & a_{01} & a_{02} \\ a_{10}...
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How $a^2(x^2-y^2)-b=0$ is a degenerate conic consisting of twice the line at infinity for $a=0$ and $b=1$?

According to Wikipedia: The conic section with equation $x^2-y^2 = 0$ is degenerate as its equation can be written as $(x-y)(x+y)= 0$, and corresponds to two intersecting lines forming an "X". ...
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On a hyperbola, find the point on the hyperbola that is distanced 3 times further from one asymptote than the other.

The equation of the hyperbola is $\ \frac{x^2}{64} - \frac{y^2}{36}=1$ So that means I have the asymptote formulas as well since I can get a and b easily out of the hyperbola equation. I then use the ...
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Rotation and translation of a conic-section

I tried to solve the following exercises, so I want to ask you if my answers are correct. 1) Given the coordinates system $(O'; X'' Y'')$ asociated to the basis $B=[{b_1=\frac{1}{\sqrt{2}}; \...
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2answers
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Rotated Ellipse (Parametric) - Determining Semi-Major and Semi-Minor Axes

Given the parametric equation $$\big(\;a \cos(\alpha+\theta), \;\;b\sin(\beta+\theta)\;\big)$$ with parameter $\theta$, how can we determine the length of the semimajor and semiminor axes, as well as ...
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Intersection of lines from lineair systems defines a conic section

I have two pencils of hyperplanes $\Sigma_1$ and $\Sigma_2$ in $P^2(\mathbb{R})$ and a projective map $\phi \colon \Sigma_1 \to \Sigma_2$. Now I need to show that the the points that are defined by $...
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Presentation on Parabola

All right, I'm presenting a class in front of some $20$ or so $12^\text{th}$ grade students in a couple of days on the topic of Parabola. I was wondering what would be the best way to teach it to them,...
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Proving Pascal's Theorem for Conics

I am trying to prove Pascal's theorem for conics. To do so, I am trying to follow the proof which is given as an exercise in Rey Casse's Projective Geometry. Here is the exercise. In this exercise, ...
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Change in basis related to conic-sections [duplicate]

I was trying to solve some change of basis exercises, so I've done them and I want you to tell me if I've done it correctly. 1) Given the coordinates system $(O'; X'' Y'')$ asociated to the ...
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Change of basis related to conics [duplicate]

I was trying to solve some change of basis exercises, so I've done them and I want you to tell me if I've done it correctly. 1) Given the coordinates system $(O'; X'' Y'')$ asociated to the basis ...
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Is $2\pi B \coth{\frac{\pi B}{2\sqrt{A^2 - B^2}}}$ a valid lower bound for the circumference of an ellipse?

Suppose there is an ellipse with a semi-major axis length of $A$ and a semi-minor axis length of $B$. The task is to prove or disprove that the circumference of the ellipse in question is greater than ...
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2answers
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The area enclosed by the locus of point C [closed]

Triangle ABC is such that AB = 4, BC = 2, and AC = 3. If vertex A is confined to the x-axis and vertex B is confined to the y-axis, what is the area of the region enclosed by the locus of all points ...
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1answer
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Dandelin sphere construction with hyperbola and ellipse intersection on same cone

When eccentricity of ellipse $\epsilon<1 $ sketch is often made in a standard disposition during proof by Dandelin with spheres inside a single nappe of cone and cutting plane touching spheres in ...
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How to show the angle is $\frac{\pi}{2}$ which between tangent of an ellipse and its chord that through the focus? [duplicate]

The proposition is told to me without proof, may it is true but I cannot to believe it with my heart before I get the proof of it or the lights. I got some problems, however, after I try to proof it. ...
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Equation of chord of a parabola whose midpoint is given

How to prove $$ T = S1 $$ $$ i.e \qquad yy_1 - 2a(x+x_1) = y_1^2 - 4ax_1=0$$ as the equation of chord for a parabola y$^2$ = 4ax whose midpoint (x$_1,y_1$) is given. $$$$ I couldn't understand how ...
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Base point of a line that intersects an ellipse

Given is an ellipse in the plane with half-axes a and b and a straight line $g$ (see image below). The line $g$ is of the form \begin{align*} g(t)=\{x+tv: t \in \mathbb{R}\}. \end{align*}, where $x$...
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1answer
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Constructing the major and minor axes of an ellipse wth compass and straightedge

Given a general ellipse (with axes not necessarily parallel to the x- or y-axes), is there a compass-and-straightedge method for constructing the major and minor axes? Just to clarify the question: ...
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Locus of right circular cone vertices resulting from constant conic intersection

Five points on $(x-y)$ plane $ (x1,y1), (x2,y2),... (x5,y5)$ defining a common conic intersection curve $C$ are obtained by section with variably inclined right circular cones i.e.,variable vertex $...
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Find parabola coefficients passing through two known points with given slopes [duplicate]

I would like to find the coefficients of a parabola passing through 2 known points. I also know the slope that the parabola should have at this 2 points. Let me show you a picture of the problem. ...
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Parametric equation for the ellipse

Let $x = A\sin(at+\theta)$ and $y = A\sin(at)$ . Prove that this parametric equation forms an ellipse except when $\theta = 0 $ and $\pi$ . My try : I expanded the $\sin(at + \theta)$ and looked for ...
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Do two ellipses with the same eccentricity, have the same distance between them all around?

enter image description here For example in the arrow shown in the photo above, assuming the two ellipses have the same eccentricity, will that distance be the same between any two parallel points ...
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Exactly 2 tangents to every non-degenerate conic in the complex projective plane

Given a point $p$ not incident to a non-degenerate conic $\mathcal{C}$ in the complex projective plane $\mathbb{C}\text{P}^2$, how would you prove that there are exactly two tangents to $\mathcal{C}$ ...
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Locating focus and semi latus-rectum of conic by R&C construction.

A cone semi-vertex angle $\gamma$ is cut by a plane inclined at angle $\alpha$ to symmetry axis. Length of perpendicular on plane from the cone vertex is $q$. It is known that eccentricity $$ \...
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Finding the area bounded by 4 parabolas [closed]

The question is: Find the area (R) bounded by the following parabolas: (1). $y=x^2$ (2). $4y=x^2$ (3). $y^2=2x$ (4). $y^2=3x$ I am looking for a solution with double integrals. I tried to do ...
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4answers
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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 ...
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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 ...
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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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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 $E'$. ...
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2answers
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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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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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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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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
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a circle and a parabola have 3 intersection points [closed]

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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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 ...
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2answers
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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}...
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1answer
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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 ...