## 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 parametric equations but the number of variables were more than the number of equations. … Read more

## Find the equation of the circle.

Find the equation of the circle whose radius is 5 which touches the circle x2+y2−2x−4y−20=0 externally at the point (5,5) Answer Hint: From the fact that the circles touch eachother externally in (5,5) it follows that (5,5) is on the line segment that connects the centers of the circle. AttributionSource : Link , Question Author … Read more

## 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? Answer By symmetry: mirror the two circles around the line that joins the centers. This leaves the circles unchanged, and so must remain … Read more

## Why is it wrong to naively pick random points inside a disk

According to MathWorld, the naive way to randomly pick points inside a disk, by using two uniformly distributed variables that are polar parameters: r∼[0,1] and θ∼[0,2π], is incorrect, leading to the distribution shown on the left. I am trying to understand why this is. It is sort of intuitive that for a given angle, points … Read more

## Displacement of a point on a circle

There is a mark on a wheel of radius $30$ cm. The mark is in contact with a horizontal plane. The wheel rotates to a distance of $10\pi \approx 31.4$ cm. $1.$ What is the angle that the old position of the mark makes with the new position? $2.$ What is the distance between the … Read more

## Find the locus of the midpoint

Find the locus of the midpoint of the chord of the circle x2+y2=a2 which subtends a 90° angle at point (p,q) lying inside the circle. I tried to solve it by taking that let the chord intersect the circle at (x_1,y_1) and (x_2,y_2). Then I found out their slopes and took their product as -1. … Read more

## The equation of a circle on a complex plane?

The equation of a circle |z−z0|=r in a complex plane has (among others) the form: z¯z+¯bz+b¯z+c=0 where b=−z0∈C. What I’d like to understand is, why is it so? Answer It’s because you can write the original form as |z−z0|2=r2 and |z−z0|2=(z−z0)(¯z−z0)=(z−z0)(¯z−¯z0) Now substitute for z0 AttributionSource : Link , Question Author : mavavilj , Answer … Read more

## Shortest distance between two circles

What is the shortest distance, in units, between the circles $(x – 9)^2 + (y – 5)^2 = 6.25$ and $(x + 6)^2 + (y + 3)^2 = 49$? Express your answer as a decimal to the nearest tenth. So I know that the first circle’s centre is at $(9,5)$ and has a radius of … Read more

## Circle inversion of a circle

Given is a circle K with radius r and centre M1. K’ is a second circle with radius r’ and centre M2 that cuts K in two points A and B so that $[M1A]$ is orthogonal to $[M2A]$ and also $[M1B]$ is orthogonal to $[M2B]$. We noted: Now through inversion on K, every circle K’ … Read more

## Problem on circles, tangents and triangles

Let c1,c2,c3 be three circles of unit radius touching each other externally. The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed by the common tangents be ΔABC. Find the length of each side of ΔABC. I don’t have a clue how to proceed. … Read more