Questions on the circle, a curve composed of points in a plane that are at a fixed distance from a fixed point.

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How to evaluate Area of $B:= \{(x,y,z) \in A | z \le 1 \}$ with $A:=\{(x,y,z) \in \mathbb{R}^3 | x^2 + y^2 = z\} = \frac{\pi}{6}(5 \sqrt{5}-1) $?

I have following problem: Let $$A:=\{(x,y,z) \in \mathbb{R}^3 | x^2 + y^2 = z\} \\ B:= \{(x,y,z) \in A | z \le 1 \}. $$ Compute the area $\mu_2(B) $. First, I thought $\mu_2(B) $ would just be the ...
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1answer
39 views

Two circles intersect in two points and the line through these two points

Consider two circles $C,C'$ in euclidean plane which intersect in exactly two points $Q,R$ and consider the line $QR$ through these points. The claim is that a point point $P$ lies on the line $QR$ ...
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1answer
41 views

How to find out how big a ball is?

Ok, This is probably a really simple question but. I need to know how I can find out how big a ball is. For example, a tennis ball is 2 1/2 inches big, but how do you find that? Though, for ...
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2answers
38 views

Is circle the only Jordan curve with this property?

When I was thinking about one problem that has to do with Jordan curves the problem which I am going to describe now, arose in my mind. And here it goes. It is known that for every $n\geq3$ the ...
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2answers
65 views

Centroid of a Triangle on a inscribed circle

$AB$ is the hypotenuse of the right $\Delta ABC$ and $AB = 1$. Given that the centroid of the triangle $G$ lies on the incircle of $\Delta ABC$, what is the perimeter of the triangle?
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0answers
62 views

Calculating the Area of a Circle Occupied by a Rectangle

This is a question regarding how to calculate the area of a circle occupied by a rectangle when that rectangle is larger than the circle (see this link for a example image ...
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6answers
68 views

How to find center of a circle given 2 points on the circle

I need a Formula for this: we have two points on the circle. How can we find the center of circle? For example $A(4.2,5.2)$, $B(5.2,6.3)$.
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4answers
32 views

get length of line connecting sector of a circle

What's the formula for getting the length of a line (in this case the red one) connecting starting point and end point of an arc, given the circle's radius R and angle A?
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4answers
65 views

Interpolation between 2 points on the perimeter of a circle?

I'm trying to produce movement on a unit circle from one point to another in equal increments, but I'm having trouble doing this without the use of angles (which isn't an option). Given 2 points on a ...
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3answers
36 views

Find the equations of the lines that pass through the point $(1,3)$ and are tangent to the circle $x^{2}+y^{2}=2$

Since the line passes through $(1,3)$ I substituted: $3=m+b$ so $m=3-b$ and $y=(3-b)x+b$. But if I then plug the line equation into the circle equation and take the discriminant, I end up with terms ...
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3answers
26 views

Find the values of b for which the line $y=3x+b$ intersects the circle $x^{2}+y^{2}=4$

I'm not sure how to tackle this question. First thought was to sketch both the line and the circle together, but it doesn't make it much clearer how to find b. Solving for both equations for y and ...
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0answers
13 views

Relations, functions and transformation (circles)

Circular ripples are formed when a water drop hits the surface of a pond. If one is represented by the equation $x^2 +y^2=4$ and then $3$ seconds later by $x^2+y^2=190$, where the length of ...
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2answers
56 views

How to calculate length of common chord of two intersecting circles?

Two circles having radii $7$cm & $19$ cm are separated by a distance of $22$ cm between their centers. If they are intersecting each other at two points $P$ & $Q$ then what will be the ...
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2answers
30 views

Evaluate $\int_C z^2 e^{1/z} \cosh(1/z)\,dz$, where $C$ is any simple-closed curve, oriented counterclockwise, and containing 0 in its interior.

Evaluate $\int_C z^2 e^{1/z} \cosh(1/z)\,dz$, where $C$ is any simple-closed curve, oriented counterclockwise, and containing 0 in its interior. my works I'm stuck in next step
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3answers
151 views

How do mathematicians invented and introduced $\pi$ term in the case of circle?

This is basic question. Since childhood I am mugging the mathematical formulae areas of square, rectangle and circle etc. Now,it is possible for me to understand formula of area of square I.e. ...
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2answers
50 views

Meaning of the expression “orientation preserving” homeomorphism

The only time that I've heard the term "orientation-preserving map" was in Linear Algebra, but today I read the term orientation-preserving homeomorphism of the circle in the following context: If a ...
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1answer
84 views

Intersection of a convex polygon and a moving circle

I have a straight line which intersects a convex polygon in 2D plane. There exists a circle with constant radius. The center of circle is moving on this line. So at first the polygon and circle don't ...
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1answer
105 views

How to find the radius of small inscribed circle?

There are three circles of radii $5 cm$, $9cm$ & $11 cm$ touching each other externally. What will be the radius of the largest circle inscribed in the region bounded by three circles? Thus ...
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2answers
86 views

Algorithm to find a line segment is passing through a circle or not?

I have a line segment between two points P1 (X1,Y1) and P2 (X2,Y2). And I have a circle at point Q(Xq , Yq) with radius R . Can I have an equation in which I can put these values and the result shows ...
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3answers
95 views

Compass-and-Straightedge Construction [closed]

I stumbled upon this question in math class, and I got stuck. The Question: You're are given a circle, and two points. How do you construct a circle that goes through the two points and is tangent to ...
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1answer
30 views

How do I convert these conics to standard form?

There are two conics I need to convert from general form to standard form but I am not sure if I am going about it right. They are $9x^2 + 5y^2 + 18x - 36 = 0$ and $2x^2 - 8x + y + 6 = 0$ The ...
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1answer
26 views

How do I plot sagitta versus arc length.

I'm working on a curvature sensor, but I'm finding it hard to find the equation for my h (sagitta) as a function of my arc length. In particular, I want to solve $\epsilon=\frac{Arc Length - Chord ...
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1answer
98 views

Area of Circle Overlapped by Rectangle

I'm trying to determine 'how much' (as a percentage) a 2D rectangle fills a 2D circle. Actual Application: I was comparing the accuracy of some computer game weapons by calculating the max possible ...
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1answer
33 views

Recurrent points and rotation number

I need to show that if $f:S^{1} \rightarrow S^{1}$ is a preserving-order diffeomorphism and $f$ has irrational rotation number, then $f$ has at least one recurrent point. How can I prove that? ...
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1answer
49 views

How to calculate new coordinates on a circle's circumference when an angle is given?

I am working on digital maps and I have a circle plotted. I have the circle's centre in lat, long and the circle's radius in meters. Now I have a point on the north end (0 deg) of this circle's ...
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0answers
41 views

Existence of a recurrent point [duplicate]

I need to show that if $f:S^{1} \rightarrow S^{1}$ is a preserving-order diffeomorphism and $f$ has irrational rotation number, then $f$ has at least one recurrent point. How can I prove that? ...
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2answers
47 views

Finding a length in a circle

A circle centred at $O$ has radius $1$ and contains a point $A$ on the circumference. Segment $AB$ is tangent to the circle at $A$ and angle $\measuredangle AOB=\theta$. If point $C$ lies on $OA$ ...
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0answers
18 views

Finding constant angular acceleration

Hello and thanks for reading my post! The question I'm working on states: Find the constant angular acceleration of an object moving along a circle if starting from rest it makes 3600 revolutions ...
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4answers
37 views

Circles and right angles

The following is a standard fact about circles: THEOREM: Let $p$ and $q$ be two antipodal points on a circle in $\mathbb{R}^2$ and let $r$ be another point on the circle such that $r \neq p,q$. ...
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1answer
58 views

Show that A is an open subset of M

If $m\in\mathbb{N}$, $M=\{0,1\}^{\mathbb{N}}$ and $A \subseteq M$ is an open set of sequence where the number 1 appears at least $m$ times. Show that $A$ is an open subset of $M$. I wanted to show ...
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0answers
35 views

Equation for the points touching a circle.

In the plane $\mathbb R^2$, a point $P$, a point $M$ and the radius $r$ are given. Suppose, that $|\overrightarrow {PM}|>r$ Then, there exist two tangents from $P$ to the circle with mid point ...
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1answer
50 views

Finding the point of intersection of two tangents to a circle

19. A circle $C$, of radius $r$, passes through the points $A (a, 0)$, $A_{1} (-a, 0)$ and $B (0, b)$, where $a$ and $b$ are positive and are not equal; a circle $C_{1}$, of radius $r_{1}$, passes ...
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1answer
38 views

Interior Angle Embedded in a Triangle Embedded in a Circle

With only knowing the angles of $B$, $C$, and $D$ (shown above), is it possible to find the interior angle $A$? And if so, how?
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3answers
48 views

The equation of a tangent to a circle at a given point

18. Show that the equation of the tangent $PT$ at the point $P \left(\frac{1}{5}, \frac{3}{5}\right)$ on the circle $$x^{2} + y^{2} + 8x + 10y - 8 = 0$$ is $3x + 4y - 3 = 0$. Find ...
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2answers
33 views

Proving a differential equation is a circle

So, I have solved the differential equation, to find the general solution of: $$\frac{y^2}{2} = 2x - \frac{x^2}{2} + c$$ I am told that is passes through the point $(4,2)$. Using this information, ...
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2answers
64 views

How to find the equation of diameter of a circle that passes through the origin?

So this was a question that I was solving that got me stuck. Its as follows: Q. Find equation of diameter of the circle $x^2 + y^2 - 6x + 2y = 0$ which passes through the origin. Now I have tried the ...
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3answers
52 views

Proof concerning circles

How do I prove that the diameter of a circle subtends a right angle at a circumference? Thank you in advance! I haven't got the slightest idea.
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0answers
42 views

To draw a perpendicular on the diameter AB of a circle from an external point P using only a straight-edge.

A perpendicular is to be dropped from external point P on diameter AB I know this question is a duplicate of potato's post, but in potatos post altitudes of triangles were used. But a property of ...
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6answers
676 views

Finding the largest triangle inscribed in the unit circle

Among all triangles inscribed in the unit circle, how can the one with the largest area be found?
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1answer
35 views

$y^2 = |\cos(\pi*x/2)|$ generates an infinite number of adjacent circles on the line $y = 0$.

http://www.wolframalpha.com/input/?i=y%5E2+%3D+%7Ccos%28pi*x%2F2%29%7C The generation for the infinite string of circles on $y = 0$. Is there a relation that generates an infinite number of square ...
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0answers
37 views

Intersecting lines in sectors of a circle.

Good day everyone, I'm trying to simulate a Laser Range Finder (LRF for short) in a corridor environment. I'm including a small fast sketch I did of this. I can't upload images yet, so I include just ...
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3answers
76 views

Given two points, how to find a circle through them that's also tangent to the $x$-axis?

A seemingly simple geometry problem that is surprisingly difficult. I want to find the radius of a circle that is tangent to the $x$-axis, but also must contain two given points. I understand there ...
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2answers
94 views

Average distance from center of circle to evenly-distributed points within it

With some number of points that are evenly/uniformly (assuming those mean the same thing) distributed within a circle of radius 1, what is the average distance from the center of the circle to a ...
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2answers
55 views

Find the length of tangent $x$.

Two circles $C_1$ and $C_2$ of radius $2$ and $3$ respectively touch each other as shown in the figure .If $AD$ and $BD$ are tangents then the length of $BD$ is $a.)3\sqrt6\\ b.)5\sqrt6\\ ...
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4answers
202 views

Find Perimeter of shaded region in semicircle. [closed]

What is the Perimeter of shaded region in semicircle if four small semicircles have radii of 1,2,3,4 respectively? a. 10 $\pi$ b. 20 $\pi$ c. 40 $\pi$ d. 60 $\pi$
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4answers
52 views

A line through the point P(8, -7) is a tangent to the circle C at the point T. Find T

Circle C equation $(x+5)^2+(y-9)^2=25$ A line through the point P(8, -7) is a tangent to the circle C at the point T. Find T. I tried simultaneous equations: 1. $(x+5)^2+(y-9)^2=25$ 2. $y = ...
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3answers
73 views

Find the area of the region $ABCD$.

In the Figure $\square PQRS$ is a square with side $2\sqrt6$. By joining the midpoints another square $\square WXYZ$ is formed . Circles are drawn with $4$ vertices as the center and radius equal ...
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4answers
44 views

A line through the point P(8, -7) is a tangent to the circle C at the point T. Find the length of PT.

Circle C equation $(x+5)^2+(y-9)^2=25$ A line through the point P(8, -7) is a tangent to the circle C at the point T. Find the length of PT. The question itself is easy when using pythagoras, ...
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3answers
50 views

How do you find the intersection(s) of two circles with equal radii? [duplicate]

I have two circles with the following equations: \begin{equation*} (x-a_1)^2+(y-b_1)^2=r^2 \\ (x-a_2)^2+(y-b_2)^2=r^2 \end{equation*} The two radii are equal. How do you find the intersections of any ...
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2answers
38 views

Help for a problem with inscribed triangles

If we have a triangle $ABC$ with $AB = 3\sqrt 7$, $AC = 3$, $\angle{ACB} = \pi/3$, $CL$ is the bisector of angle $ACB$, $CL$ lies on line $CD$ and $D$ is a point of the circumcircle of triangle $ABC$, ...