Let C be the circle with center at (1,1) and radius 1. If T is the circle centered at (0,y) passing through origin and touching the circle C externally, then the radius of T is equal to

Solution :

Let coordinates of the center of T be (0, k).

Distance between their center is

k + 1 = \(\sqrt{1 + (k – 1)^2}\)

where k is radius of circle T and 1 is radius of circle C, so sum of these is distance between their centers.

\(\implies\) k + 1 = \(\sqrt{k^2 + 2 – 2k}\)

\(\implies\) \(k^2 + 1 + 2k\) = \(k^2 + 2 – 2k\)

\(\implies\) k = \(1\over 4\)

So, the radius of circle T is k i.e. \(1\over 4\)