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

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Help with this coordinate geometry question involving cirlces and parabolas.

Question: A point $P$ in a plane moves such that it remains at a fixed distance $r$ from a fixed point $A\equiv(r,r)$. (i) Find the equation of the locus of point $P$ (in terms of $r$). Another ...
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0answers
27 views

How to get points of lines tangent to 2 circles

How do you get the points of lines tangent to two circles, where both starting point and end point of line are exactly touching their respective circles? Please help. Thanks
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2answers
47 views

How to calculate radius of a spherical surface having four circles touching one another?

There are four circles having radii $r_1, r_2, r_3 $ and $r_4$ touching one another on a spherical surface of radius $R$ (as shown in the picture below, four colored circles touching one another at 6 ...
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1answer
30 views

Find point inside circle but outside of n- other circles

There is one green circle and 0 to n red circle(s). I'm trying to find a point inside the green circle, but outside all red ...
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1answer
64 views

Find circumcenter when distance between ABC points of triangle with two points's ratio given

The complete problem is: I am having three points A,B,C whose ratio of the distances from points (1,0) and (-1,0) is 1:3 each. Then I need the coordinates of the circumcenter of the triangle formed ...
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1answer
72 views

What are the distances from a line to the tangents of a circle?

I have a line given by two points, and a circle given by its origin and radius. I need to find the perpendicular distance between the line and the two tangents of the circle that are parallel to the ...
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2answers
29 views

Interpolate/Increment Vector Rotation

For my 2D physics engine, I'm using the unit vectors of the direction an object is facing to represent its orientation; essentially, [Cos(theta),Sin(theta)] where theta is the object's rotation in ...
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1answer
20 views

Central angle of a circular sector from area and arc length

I've been doing a task which says the following: Area of a circular sector is $3.2\pi cm^2$, arc length is $0.8\pi cm$. What is the central angle? I've been attacking this from several angles ...
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1answer
40 views

Circle passing through intersection points of two bigger circles

Suppose the equations of two intersecting circles are given.Now how to find the equation of circle passing through the points of intersection of the larger circles? Now please dont tell me that i got ...
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2answers
31 views

Find a line parallel to a known line that intersects a known circle at one point.

There is a circle with an equation $x^2+y^2=16$ and a line with equation $y=x+1 $. The question is to find an equation of line placed parallel to this line and touching the circle at only one point. ...
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1answer
36 views

Circular motion of a particle with increasing speed.

I have researched angular acceleration and circular motion on google, but haven't found what I am looking for. I hope you can help me find more information about the problem below, with particular ...
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2answers
65 views

geometry question-semicircle inscribed in quadrilateral.

there is a line AD whose midpoint is O (AO=OD). a semicircle is drawn with centre O and any radius < AO with its straight edge being part of line AD. lines AB and CD are drawn tangent to the ...
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61 views

Writing circles as $|z-a| = \lambda |z-b|$ for the same $a,b$

My problem is in the context of the complex plane. I want to know if given two disjoint, not concentric circles $C_1,C_2\subset \mathbb{C}$, can you find $a,b\in \mathbb{C}$ such that $$C_1=\{z\in ...
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1answer
61 views

BMO1 2009/10 Question 4 Geometry Problem

Two circles, of different radius, with centres at B and C, touch externally at A. A common tangent, not through A, touches the first circle at D and the second at E. The line through A which is ...
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1answer
23 views

converting a circle's equation not touching axis to polar from Cartesian system for integration

I am having a really hard time figuring out how to convert this circle to polar coordinates, I am to use double integration after converting it. I know that $\theta$ has to be between $0$ and $\frac ...
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0answers
23 views

Calculate the overall circle enclosing multiple smaller circles

I have multiple smaller circles of a fixed radius that I am using to define a larger enclosing circle. So I'll need to find the x and y and radius of this new circle. I am looking for efficient over ...
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1answer
24 views

kth root of an open set in the circle toplogical group

My intuition tells me that in the topological group of the circle, if I take an open set U, then its kth root (where k is some natural number) in the circle is also an open set. In order to show it I ...
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0answers
10 views

Cover a rectangular with circles with different radius

I have a question about covering a rectangular with a set of circles, which is similar to this problem. I have a rectangular, say 1 meter by 2 meters. The whole rectangular is segmented horizontally ...
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1answer
63 views

Gradient function of a circle

The parametric equations of a circle $C$ are: \begin{align*} x&=2+\dfrac{13}{5\sqrt{2}}\cos t\\ y&=1+\dfrac{13}{5\sqrt{2}}\sin t \end{align*} for $t\in[0,2\pi]$. I am stuck on this part: Find ...
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1answer
105 views

Writing 1+1= 2 in a complicated way

I am learning Unit Circle at the moment and I am using this source as an education tool Trigonometry: Unit Circle (Starts at 20:00). The author solves these simple equations like below: ...
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2answers
60 views

Length and breadth of a rectangle enclosed between two semi-circles of given radii

34. It is required to take a rectangular frame in a horizontal position along a corridor bounded by vertical walls of which a horizontal cross-section is two concentric semicircles of radii $r$ and ...
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0answers
76 views

Simple Math\Geometry Question about Circles, etc

I never took Geometry in school, and although I went all the way to stats in college, all the brain surgeries I had made it really hard for me to do the simplest math for some unknown reason. So I'm ...
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3answers
68 views

Different ways to prove that area of circle with radius $r$ is $A=r^2\pi$

I want to compute the area of a circle in different ways. I know that any circle with radius $r$ have area $A=2\int_{-r}^r\sqrt{r^2-x^2}dx=r^2\pi$, but I want to prove it in other ways. My first way ...
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1answer
24 views

flat approximation of a circle at a point

I need to find a flat approximation of a circle at a given point. The circle I am working with is $$x^2+y^2=\frac9 4$$ The point is $(1,\sqrt{\frac 5 4})$ I have found an approximation, but it is ...
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2answers
71 views

Maximum area of a rectangle

Two concentric circles have radii 13 and 15. Let ABCD be a rectangle, so that A and B lie on the larger circle, and C and D lie on the smaller circle. Find the maximum area of rectangle ABCD. I tried ...
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1answer
40 views

Tangent - point of contact [closed]

I want to find out the co-ordinate of point of contact of tangent to a circle from external point when its center and radius are known. Please Help
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1answer
31 views

21' round pool in a square.

How big of a square would I need to fit a 21'round pool in with an extra foot on all sides.I was 23' or 24'.I also thought the radius plus a foot. However it has been 6 years since my last college ...
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4answers
252 views

Locus of a midpoint

Let $Γ_1$ be a circle of radius $4$, and let $Γ_2$ be a circle of radius $14$. The distance between the centers of $Γ_1$ and $Γ_2$ is $25$. Let $A$ be a variable point on $Γ_1$, let $B$ be a variable ...
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1answer
55 views

Length from tangent circles

A circle $Γ_1$ of radius $25$ is externally tangent to a circle $Γ_2$ of radius $16$ at $C$. Let $AB$ be a common direct tangent, so that $A$ lies on $Γ_1$ and $B$ lies on $Γ_2$. Draw the tangent to ...
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2answers
39 views

intersection points of two circles

I am trying to find the points at which two circles intersect. The circles I am working with are: $$(1) \qquad x^2+y^2 = \frac{9}{4}$$ $$(2) \qquad (x-2)^2+y^2=\frac9 4$$ I am following this answer: ...
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4answers
227 views

What is the relationship here?

This is an annoying and probably easy question. How does one solve and approach it?
166
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27answers
30k views

Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers

An exam for high school students had the following problem: Let the point $E$ be the midpoint of the line segment $AD$ on the square $ABCD$. Then let a circle be determined by the points $E$, $B$ and ...
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1answer
78 views

2n points with arrangement on circle?

I Read a Short Questions on Mathematics and get stuck in one challenging problem. any idea from my friends? Suppose 2n points with arrangement ...
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3answers
119 views

How can the centers of these 5 related circles be specified as a formula?

This is my first time posting in this forum, so please forgive me if my question is too involved or if I've posted it in the wrong area. I hope someone finds it interesting enough to try their hand at ...
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2answers
92 views

How many sides can a polygon have before it is “considered” a circle?

Good day, my family had a dinner discussion about polygons and how many sides a polygon has in relation to the angle measurement you'll get when you measure an "arc" encompassing a "side" of the ...
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1answer
8 views

closure of dual group in pointwise convergence.

I have something which seems kind of trivial but I can't seem to prove it. Let G be an abelian topological group and let T be the circle group. Denote by G* the group of all continuous homomorphisms ...
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3answers
45 views

What is the equation of a general circle in 3-D space?

I know that $(x-x_0)^2+(y-y_0)^2-r^2=0$ is a general planar circle and $(x-x_0)^2+(y-y_0)^2+(z-z_0)^2-r^2=0$ is a general sphere. I want to know the general expression of a circle in space. Can ...
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1answer
93 views

Find the area of the shaded region

This is the Figure, $ABCD$ is a square , $AB = BC = CD = DE = 21cm$. $AC$ and $BD$ are the diagonals of the square. The two semi circles are drawn with $AD$ and $BC$ as diameter. Find total are of ...
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1answer
33 views

How to solve a ratio between radiuses of the attached diagram?

I have a red circle with a radius of 1. I'd like to get the exact ratio between it and the bigger blue circle radius. So how do you calculate AI/AJ in this case? I suppose it going to involve square ...
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2answers
43 views

Standard form of the equation of a circle : does the order matter?

The distance formula is $d = \sqrt{(x2 - x1)² + (y2 - y1)²}$ If the two points are (1, 2) (3, 4), it doesn't matter whether one write : $d = \sqrt{(1 - 3)² + (2 - 4)²}$ Or $d = \sqrt{(3 - 1)² + ...
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1answer
21 views

Polar Represantation of Shifted Disk

How to represent a shifted circle or disk (I mean the center of the circle is not at origin) in polar coordinate? For example I have a circle/disk in z-Domain like this: I thought this: $z = ...
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2answers
74 views

Geometry Problem about Circles and Tangents

It is the second problem from my maths notebooks, which is still unsolved. I translated it from Russian, so their may be some discrepancies in translation. So, I also added image. First problem was ...
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1answer
88 views

Find the Center of Tangent Circles

One of my math tests has this question. A circle has its center at $(6,7)$ and goes through $(1,4)$. Another circle is tangent at $(1,4)$ and has the same area. What are the possible ...
4
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1answer
38 views

Fit circle between points located on unit-sphere

Suppose I have a sphere of points with two coordinates (two angles), all points are located on a unit sphere, so radius of the sphere is one. Now my problem is, I want to find empty circles, or ...
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2answers
47 views

Find the side of the pentagon

Let $ABCDE$ be a pentagon inscribed in a circle. Given that $AB=CD$, $BC=2AB$, $AE=1$, $BE=4$, $CE=8$. Find $DE$. I am unable to use the properties of circle in this question. All I could do was fund ...
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5answers
44 views

The Equation of a Tangent Line to a Circle at a Point

How do you determine the equation of the tangent line to the circle $(x-4)^2 + (y+3)^2 = 25$ at $P(8,-6)$? Thank you in advance.
3
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4answers
63 views

Limit of ratio of areas of triangles defined by tangents to a circle

Let $AB $ be an arc of a circle. Tangents are drawn at $A $ and $B $ to meet at $C $. Let $M $ be the midpoint of arc $AB $. Tangent drawn at $M $ meet $AC $ and $BC $ at $D $, $E $ respectively. ...
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1answer
41 views

Equation of a circle with a center that passes through a point [duplicate]

Write the equation of the circle with center at (-1,3) and passes through the point (2,-4).
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1answer
54 views

What is the probability of a circle sector being chosen at random?

What should I do in order to find the probability of the sector? Can you please give me the answer to this. It will really help me out if you do.
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5answers
279 views

How do I find the area of this shaded circle?

This circle has been bugging me for a while, and I do not know how to solve it. Can someone help me find the area of the shaded circle? It would really help me.