Thread: Circles

The circle with center and radius touches externally the circle center radius . The tangent at their common point passes through the origin. Show that,

If also, the other two tangents from the origin to and are perpendicular, prove that .
Hence show that if remains fixed but and vary, then lies on the curve
.
I'm quite sure if I know how to do the first part, I should be able to do the rest, but I don't even know how to start. Thanks for any help.

Arze: "The circle with center and radius touches externally the circle center radius . The tangent at their common point passes through the origin. Show that,
"

Solution for (i) if they have a common point of (0,0), use the standard formula for circle:

(0 - A)^2 + (0 - B)^2 = R^2,
(0 - a)^2 + (0 - b)^2 = r^2,

subtract,

(A^2 - a^2) + (B^2 - b^2) = (R^2 - r^2)

but how do we know that the point is (0,0)?