# Tagged Questions

For questions conserning circles. A circle is a curve composed of points in a plane that are at a fixed distance from a fixed point.

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### Probability of being in a circle, given normal

Let's assume a bivariate normal distribution with center $\mu$ and covariance matrix $\Sigma$. Let a circle $C$ be given as $C=\{x\in\mathbb{R}^2:||x-\mu||\leq R\}$. I would like to calculate the ...
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### Find area of this circle

Given the circle O with perpendicular diameters and a chord, find the area of the circle if $EF = 8"$ and $DE = 20"$ inches.
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### The intuition behind choosing this length?

In Problem 1, RMO 2004 there is a particular choice of length which leads to the solution, the length being that of the tangent from the foot of the perpendicular to the circle. Just a rundown of ...
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### how is it that $\int_0^x 2\pi r\ dr$ is equal to the area of a circle [closed]

I'm studying calculus and I'm having some basic questions, this one is regarding the area of a circle. we know, from some guy, that the circumference of a circle is $2 \pi r$ and the area can be seen ...
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### Proving circumcenter lies on altitude

Problem: In $\triangle ABC$, let $D$ be the intersection of the tangents to the circumcircle at $B$ and $C$, let $B'$ be the reflection of $B$ across $AC$, let $C'$ be the reflection of $C$ across $AB$...
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### How to calculate the shortest rotation from current to the target angle? [closed]

In the following situation: My current angle is 40*, my target angle is 130*. How should I calculate the rotation that should be done to reach the target angle from the current one? I've done the ...
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### Locus of intersection point of perpendicular tangents

Here is the question which I am referring to:A tangent is drawn to the circle $(x-a)^2+y^2=b^2$ and a perpendicular tangent to the circle $(x+a)^2+y^2=c^2$,find locus of their point of intersection. ...
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### Why Circle is traced counterclockwise and ellipse is traced clock wise?

In the Lecture 32: Polar Coordinates,professor traces the circle counterclockwise, but traces the ellipse clockwise. "Which was this one here. And first we noted that this does parameterize, as we ...
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### What happens when $r \to \infty$? Will it be a line? (partial circle)

Let $a$ be a arc of particle circles, which is constant. What happens when $r \to \infty$? Will it be a line? Radius of partial circle : $r$, Arc of partial circle : $a$ and constant, For $r=r_0$ ...
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### How to check two circles are linked or not? (without using topology)

In $\mathbb{R}^6$, three loops $$C_1:=\{(0,x,-x;0,y,-y)\mid x^2+y^2=1\}\\ C_2:=\{(x,0,-x;y,0,-y)\mid x^2+y^2=1\}\\ C_3:=\{(x,-x,0;y,-y,0)\mid x^2+y^2=1\}$$ are embedded. Is there a pair of circles ...
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### Determining Locations of Circles to Optimally Cover a Polygon

I want to completely cover a region on a map(Continental US)/polygon with circles of a certain radius. Is there a way to determine the best locations and how many circles would be needed to completely ...
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### How to calculate $\Delta$ in conic sections?

When learning conic section I learnt about $\Delta$. For any conic in general form : $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ Here $\Delta=abc +2fgh - af^2 - bg^2 -ch^2$ The conic is said to be ...
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### Radius of inner circles given radius of outer circle and number of inner circles in circular fractal

I am trying to create a circular fractal in which each circle is composed by a given number $n$ of smaller circles. It would look something like this for $n = 8$: However, I don't know how to ...
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### How to determine if the implicit curve is closed?

Let the implicit equation $$F(x,y)=0, \quad (x,y)\in\mathbb{R}^2$$ defines a curve $\gamma$. The question is what properties must have the function $F$, s.t. the curve $\gamma$ be topologically ...