question_answer

Two circles of radii 9 cm and 2 cm respectively has centres X and Y and\[\overline{XY}=17\,\,cm.\]Circle of radius r cm with centre Z touches two given circles externally. If\[\angle XZY=90{}^\circ ,\]then find r.            [SSC (CGL) 2012]

A) 18 cm              

B) 3 cm

C) 12 cm              

D) 6 cm

Correct Answer: D

Solution :

In \[\Delta XYZ,\]By Pythagoras theorem,

\[\therefore \]\[X{{Y}^{2}}=X{{Z}^{2}}+Z{{Y}^{2}}\]

\[\Rightarrow \]\[{{17}^{2}}={{(9+r)}^{2}}+{{(r+2)}^{2}}\]

\[\Rightarrow \]\[289=81+13r+{{r}^{2}}+{{r}^{2}}+4r+4\]

\[[\because {{(a+b)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab]\]

\[\Rightarrow \]   \[2{{r}^{2}}+22r-204=0\]

\[\Rightarrow \]   \[{{r}^{2}}+11r-102=0\]

\[\Rightarrow \]   \[{{r}^{2}}+17r-6r-102=0\]

\[\Rightarrow \]\[r\,\,(r+17)-6\,\,(r+17)=0\]

\[\Rightarrow \]   \[(r-6)(r+17)=0\]

\[\Rightarrow \]               \[r=6\,\,cm\]