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

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

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

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