Let C be the circle with centre at (1,1) and radius 1. If T is the circle centred at (0,y), passing through the origin and touching the circle C externally, then the radius of T is equal to:
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Official solution
Equation of circle \[C \equiv {\left( {x - 1} \right)^2} + {\left( {y - 1} \right)^2} = 1\]
Radius of \[T = \left| y \right|\]
T touches C externally
therefore,
Distance between the centers = sum of their radii
\[ \Rightarrow \sqrt {{{\left( {0 - 1} \right)}^2} + {{\left( {y - 1} \right)}^2}} = 1 + \left| y \right|\]
\[ \Rightarrow {\left( {0 - 1} \right)^2} + {\left( {y - 1} \right)^2} = {\left( {1 + \left| y \right|} \right)^2}\]
\[ \Rightarrow 1 + {y^2} + 1 - 2y = 1 + {y^2} + 2\left| y \right|\]
\[2\left| y \right| = 1 - 2y\]
If y>0 then 2y=1−2y \[ \Rightarrow y = {1 \over 4}\]
y<0 then −2y=1−2y \[ \Rightarrow 0 = 1\] (not possible)
\[\therefore\] \[y = {1 \over 4}\]