Question: For the following pairs of curves, Decide whether there exists an inversion that transforms one onto the other. Identify the inversion if it exists.
- The circles $x^2+y^2=16$ and $(x-34)^2+y^2=900$
The first circle with the equation $x^2+y^2=16$ shall be call $C$. $C$ is centered at the origin $(0,0)$ and has a radius of $4$.
The second circle with the equation $(x-34)^2+y^2=900$ shall be call $D$. $D$ is centered at $(34,0)$ and has a radius of $30$.
That is the amount of information I was able to obtain by my general knowledge of circles. Now, if anyone can help me determine how to know if there exist an inversion?
Your setup is symmetric about the $x$ axis. So your circle of inversion has to have its center on that axis as well. Furthermore it has to map the pair of points where $C$ intersects the $x$ axis onto the pair where $D$ intersects the axis. Which means
An inversion is an involution, so applying the inversion twice will give the identity transformation. This means the first alternative is not an option, and the second one is the one you need. So you need a circle of inversion with center $(x,0)$ and radius $r$ which satisfies
$$(x+4)(x-64)=(x-4)^2=r^2$$
I believe you can take it from here.