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.

2,993 questions
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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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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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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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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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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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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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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?