Given a locus of points, show that it is a circle (find its radius and center)

[Goomba]

1. Suppose L is the line y=a. Show that the locus of points {1/z: z E(epsilon) L} is the circle of radius 1/(2a) with center at -i/(2a).



2. T(z)=(az+b)/(cz+d), ad != bc. T(z) is a linear fractional transformation. Linear fractional transformations take circles to circles, where the class of circles includes regular circles and straight lines.

Equation for circle on the rectangular plane: (x-a)^2+(y-b)^2=r^2 where r is the radius and (a,b) is the center.

Equation for circle on the complex plane: |z-center|=radius, I think??



3. The preceding problem with my solution:
Let C be the circle |z|=r. Show that the locus {1/z: z E(epsilon) C} is another circle with center at the origin. Find its radius.

Let z=a+bi.
Given: |z|=r
r=sqrt(a^2+b^2)

|1/z| = |1/(a+bi)| = |(a-bi)/(a^2+b^2)| = |a/(a^2+b^2) - ib/(a^2+b^2)| = sqrt{[a/(a^2+b^2)]^2 + [-b/(a^2+b^2)]^2} = sqrt[(a^2+b^2)/(a^2+b^2)^2] = sqrt[1/(a^2+b^2)] = 1/r, which is its radius.

I don't know how to show its center as the origin. Or even if I correctly found the radius here. The whole process of finding center and radius given a locus of points confuses me...

 

[HallsofIvy]

1. Suppose L is the line y=a. Show that the locus of points {1/z: z E(epsilon) L} is the circle of radius 1/(2a) with center at -i/(2a).



2. T(z)=(az+b)/(cz+d), ad != bc. T(z) is a linear fractional transformation. Linear fractional transformations take circles to circles, where the class of circles includes regular circles and straight lines.

Equation for circle on the rectangular plane: (x-a)^2+(y-b)^2=r^2 where r is the radius and (a,b) is the center.

Equation for circle on the complex plane: |z-center|=radius, I think??



3. The preceding problem with my solution:
Let C be the circle |z|=r. Show that the locus {1/z: z E(epsilon) C} is another circle with center at the origin. Find its radius.

Let z=a+bi.
Given: |z|=r
r=sqrt(a^2+b^2)

|1/z| = |1/(a+bi)| = |(a-bi)/(a^2+b^2)| = |a/(a^2+b^2) - ib/(a^2+b^2)| = sqrt{[a/(a^2+b^2)]^2 + [-b/(a^2+b^2)]^2} = sqrt[(a^2+b^2)/(a^2+b^2)^2] = sqrt[1/(a^2+b^2)] = 1/r, which is its radius.

I don't know how to show its center as the origin. Or even if I correctly found the radius here. The whole process of finding center and radius given a locus of points confuses me...

You found the radius by taking |1/z|, that is, the distance from the point 1/z to the origin. In other words, you have shown that the distance from any point to the origin is a constant. Doesn't that show that the locus is a circle with the origin as center?