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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Parametric Equation of conics: Parabola

Let $P(ap^2,2ap)$ and $Q(aq^2,2aq)$ be two points on the parabola $y^2=4ax$ such that PQ is the focal chord. Let $A(at^2,2at)$ and $B(as^2,2as)$ be two other variable points on $y^2=4ax$. a) Show ...
2
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3answers
49 views

Proof related to circle

How can I prove that if two circles, one entirely inside the other, intersect at a point, then that point of intersection must be collinear with the centers of the two circles?
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3answers
62 views

find equation of the middle circle

The diagram below is of 3 circles have 3 centres A, B and C and they are collinear. The equations of the circumferences of the outer circles are ${(x + 12)^2 + (y + 15)^2 = 25}$ and ${(x - ...
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1answer
78 views

Maximum number of pythagorean triples on a circle not centered on the origin

Suppose we two equations $$x^2+y^2=r^2$$ and $$(x-a)^2+(y-b)^2=2g^2$$ Where x,y and r are integer variables greater than 0. a,b and g are integer constants greater than 0. I conjecture that for any ...
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2answers
41 views

Finding points in circles

so I have two questions Im stuck on and I really do not know what to do at all. Thank you. 1) 2)
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2answers
17 views

Find the equation of the tangents given the gradient

We're given that $x^2 + (y+2)^2 = 4$ and we're asked to find the equation of the lines where the gradient $=1$ Through implicit differentiation I got $x + (y+2)y'=0$ and if $y'=1$ then: $y=-x-2$ is ...
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1answer
35 views

Equation of circle in 3d Plane?

Suppose I have a sphere centered at origin. $$ x^2+y^2+z^2=5 $$ and a plane $$ \vec{r}.(\hat{i}+\hat{j}+\hat{k})=3\sqrt{3} $$ And this plane cuts the sphere at a circular region. How do I write the ...
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1answer
32 views

Finding the length of the line segment $JI$

In the diagram, $J$ is the circumcenter of $\Delta ABC$ and $I$ is the midpoint of $BC$. How can i show that $JI=\cfrac {R}{ \cos A}$ ? I simple don't know how to prove such simple fact...
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1answer
55 views

Area at centre of Venn circles

How much information do we need to calculate the area of the centre of $3$ Venn circles? I would guess we need to know the lengths of the sides of the triangle formed by the circle centres and the ...
3
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2answers
99 views

How to find the area of a square inside a semicircle using only the radius?

Provided with only the radius of the semicircle (10 cm) and the knowledge that the corners of the square touch the semicircle, how can one find the area of this square?
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0answers
48 views

Show that a complex equation represents a circle

I'm having troubling understanding the answer to a question. The question is: If $\ v=1+i$ and $\ z=x+iy$, for any real numbers x and y: Show that the equation $\left|z-v\right|= \left|vz\right|$ ...
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0answers
49 views

Circle continuty principle proof

Circular continuity principle: If a circle C has one point inside and one point outside another circle C' , then the two circles intersect in two distinct points. I read this on Euclidean and ...
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0answers
25 views

Length of an arc of a circle when the angle is infinitesimally small

The task is to express the length of an arc of a circle trapped between two radii named $r$ if the angle between them is infinitesimally small, named $d\theta$. The solution to this problem is ...
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1answer
23 views

Scaling intersecting circles that are locked to individual rays

I have two circles that intersect and I need to find the scalar value where they are only touching at one point. Each circle is locked to a point on it's circumference and can only scale relative to ...
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1answer
33 views

Hyperbolic space and metrics

Using metrics is it possible to derive the circumference and area of a circle in hyperbolic space. I've found that the answer (without using metrics) are: C=2πsinh(r) and A=4πsinh2(r/2). But I'm ...
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1answer
48 views

Angle of point on one circle to match view from another circle

This should be a simple geometry problem, but I can't seem to find a simple answer.
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2answers
59 views

Given locus is a circle, prove two lines are perpendicular

Let $l_1$ and $l_2$ be two lines in the plane. The locus of all points $P$, such that the sum of squares of the distances of $P$ to $l_1$ and $l_2$ is constant, is a circle. Prove that $l_1$ and $l_2$ ...
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1answer
32 views

When is this conclusion true?

Assume $A,B,C,D$ are arcs in the a circuit of radius $R=1$, all $A,B,C,D \le \pi$. What is/are the main constraint(s) for the following conclusion to be correct? $$ A+B < C+D \implies ...
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0answers
25 views

Complex Numbers using relations

A straight line $T$ passes through $p=\sqrt{3}-2+3i$ and is a tangent to $S$, where $S$ is defined by $S=\left\{ z:\; \left| z+2-2i \right|=2,\; z\in C \right\}$. $T$ can be defined by $T=\left\{ ...
2
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1answer
105 views

Equation of a circle in matrix form

I have an equation $ \left( x-3 \right)^{2}+\left( y-3 \right)^{2}=9 $, and am trying to apply a matrix rotation of 180 degrees to it, however, I am having difficulty transferring the equation of the ...
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1answer
26 views

Point of intersection between two circles how do I get the point?

Circle1 with $(1,1)$ and $r=1$ Circle2 with $(3,2.5)$ and $ r=2$ Best way to calculate the intersection without a calculator on a piece of paper, I tried many ways which I saw on the internet and ...
2
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2answers
26 views

Constant of a hyperbola

Hyperbolas are a companion to a circle, sharing many properties when it comes to their trig functions and equation. But, if the circle has $\pi$ as a constant relation, does a hyperbola have some ...
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3answers
27 views

Can a circle be specified by an arbitrary triangle when one of its sides and the angle opposite to it is known?

Given an arbitrary triangle where one of its sides is $a$ and the angle opposite to it is $A$, is there a circle with a unique radius $r$ such that this triangle is inscribed within it?
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2answers
55 views

Find the line with a positive slope that is tangent to two circles

There is a line with positive slope which is tangent to the circle x^2+y^2=1 at some point P and which is also tangent to the circle $(x-3)^2 + y^2 = 4$ at a point Q. Find the equation of this line.
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0answers
39 views

Maximum regions chords can divide a circle

What is the maximum number of regions into which n chords can divide a circle? I have gotten all my data, and I am having trouble with writing the equation. I notice that the first difference is ...
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3answers
136 views

Find the area and perimeter of a segment of a circle

Find the area and perimeter of the shaded region in the figure My work $$ \begin{align} \text{Area} &= \frac{r^2}{2} \theta - \frac12 r^2 \sin(\theta)\\ &= \frac{8^2}{2} \frac{37 ...
2
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1answer
33 views

Integral Apollonian circle packing with unique curvatures

I was wondering if it is possible to construct an Apollonian gasket where every circle has a unique integer curvature. Take for instance the following gasket, defined by curvatures (−10, 18, 23, 27): ...
2
votes
1answer
64 views

Double integration in polar coordinates between two circles

I am trying to integrate converting to polar coordinates, between two circles. $$A = \iint_D x \,\mathrm{d}A $$ Ant the domain of integration is set to be the region in the first quadrant between ...
2
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2answers
25 views

Product of inscribed circle and circumscribed circle radiuses [closed]

Let a and b be the two shortest sides of a triangle. r is the radius of the inscribed ...
1
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1answer
31 views

Is a shape formed of two tangents and radii symmetrical?

Is the kite formed by the two tangents and radii in this image symmetrical? Is there a law or reason why? I am assuming that the two tangents are of equal length, but I can't see why. Are any two ...
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2answers
190 views

How to avoid overlap in circle fractals?

I had asked this on reddit and someone suggested that I try here: Assuming that the pattern in the image below continues infinitely, how much would each generation of circles have to decrease to ...
0
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1answer
34 views

Solving geometric problem

I want to find the coordinates of the $p$ point and $\beta$ angle in the following figure. The point is defined by the angle $\alpha$, the positions of the $a$ point, and the radius of the circle $r$, ...
2
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2answers
83 views

Circles tangent to a parabola

For the past two weeks I was struggling with solving the following problem. Description of variables: $(x_n,y_n)$ - center point of the circle $C_n$ $r_n$ - radius of the circle $C_n$ Given the ...
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1answer
28 views

Suppose there exist exactly $n$ circles with non-zero radius in the plane tangent to all the three lines,then the possible values of $n$ is/are

Three distinct lines are drawn in a plane.Suppose there exist exactly $n$ circles with non-zero radius in the plane tangent to all the three lines,then the possible values of $n$ is/are ...
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2answers
52 views

Do two circles always have a radical axis?

Do two circles always have a radical axis? I came across this question in my book.I think every pair of circles have a radical axis,but my book answer says NO,not every pair of circles have a ...
1
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2answers
67 views

Find the equation of the circle which cuts the circle $x^2+y^2+2x+4y-4=0$ and the lines $xy-2x-y+2=0$ orthogonally

The equation of the circle which cuts the circle $x^2+y^2+2x+4y-4=0$ and the lines $xy-2x-y+2=0$ orthogonally,is ...
0
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0answers
59 views

If two circles passing through $P$ touch coordinate axes and also cut at right angles,then

If $P(a,b)$ is a point in the first quadrant.If the two circles which pass through $P$ and touch both the coordinate axes and also cut at right angles,then: ...
2
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1answer
54 views

Radius of circle [closed]

A circle is tangent to X and Y axis and a straight line given by $y=-\sqrt{3}x+5\sqrt{3}+15$. Then what is the radius of circle? Can it be determined by this info?
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2answers
34 views

Number of points $C=(x,y)$ on the circle $x^2+y^2=16$ such that the area of the triangle whose vertices are $A,B$ and $C$ is a positive integer

Let $A(-4,0)$ and $B(4,0)$.Number of points $C=(x,y)$ on the circle $x^2+y^2=16$ such that the area of the triangle whose vertices are $A,B$ and $C$ is a positive integer,is... I found the area of ...
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3answers
32 views

A straight line $l_1$ with equation $x-2y+10=0$ meets the circle with equation $x^2+y^2=100$ at $B$ in the first quadrant.

A straight line $l_1$ with equation $x-2y+10=0$ meets the circle with equation $x^2+y^2=100$ at $B$ in the first quadrant.A line through $B$,perpendicular to $l_1$ cuts the $y-$axis at $P(0,t)$.The ...
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1answer
43 views

The equation of the image of the circle $x^2+y^2+16x-24y+183=0$ by the line mirror $4x+7y+13=0$ is

The equation of the image of the circle $x^2+y^2+16x-24y+183=0$ by the line mirror $4x+7y+13=0$ is: $(A)x^2+y^2+32x-4y+235=0$ $(B)x^2+y^2+32x+4y-235=0$ $(C)x^2+y^2+32x-4y-235=0$ ...
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0answers
12 views

Locus of keeping distance with another moving object

A submarine at point $(0,0)$ travelling at max speed of $S_u$ and a ship at point $(d,0)$ traveling at $S_h$ with $f(t)=S_ht$ where $S_u<S_h$. The submarine try to ambushing the ship ahead of the ...
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3answers
250 views

Point circle from pair of straight line equation

In general, $$x^2+y^2+2gx+2fy+c=0$$ represents a circle with centre at $C(-g,-f)$. Equations of the form $$ax^2+2hxy+by^2=0$$ represents a pair of straight lines passing through origin. But the ...
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1answer
15 views

Geting the unit vector of a tangent at point x/y on a cricle

Trying to write an programming algorithm but having a couple issues with the maths. So I have an object moving in circlular motion in one plane. The circle is defined with a center co-ordinate and a ...
28
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3answers
476 views

Can all circles of radius $1/n$ be packed in a unit disk, excluding the circle of radius $1/1$?

This problem occurred to me when I came across a similar problem where the radii were taken over only the primes. That question was unanswered, but it seems to me infinitely many circles of radius ...
0
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1answer
34 views

Length of a line segment between arcs of two circles on line of centers

In the following image, I'm trying to solve for $x$ in terms of $r$, $s$, and $\theta$. This problem turns up in a non-linear geometry control theory proof I'm writing. However, sadly, I've ...
2
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3answers
53 views

Pythagorean theorem vs an equation of a circle?

Today I told my teacher that the equation of a circle looks like to the Pythagorean theorem to me, but he said that I'm wrong and to re think it. Why $(x-h)^2 + (y-k)^2 = r^2$ is not a PT, it looks ...
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1answer
42 views

Find the circle circumscribing a triangle related to a parabola [duplicate]

Consider the following lines $x-y-1=0$ $x+y-5=0$ $y=4$ The line 1 is the axis of the parabola, the line 2 is the tangent at the vertex to the same parabola, and the line 3 is another tangent to ...
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1answer
32 views

Solving double integral, sections of a circle

New here, so I'll try get this asked right the first go. I have a double integral I need solved. I actually have the solution, and have verified it myself through numerical integration, but I would ...
0
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1answer
16 views

Any higher level maths or theories for epicycloids and/or hypocycloids?

For my 12 grade folio task on cycloids, I need to research hypocycloids and/or epicycloids. I need to consider: - exploring how the relative radii of the circles relate to the path - develop ...